On the afternoon of the 3d of July a letter was received from Lieutenant-General Pemberton, commanding the Confederate forces at Vicksburg, proposing an armistice and the appointment of commissioners to arrange terms for the capitulation of the place. The correspondence, copies of which are herewith transmitted, resulted in the surrender of the city and garrison of Vicksburg at 10 o'clock a.m., July 4, 1863, on the following terms: The entire garrison, officers and men, were to be paroled, not to take up arms against the United States until exchanged by the proper authorities; officers and men each to be furnished with a parole, signed by himself; officers to be allowed then side arms and private baggage, and the field, staff and cavalry officers one horse each; the rank and file to be allowed all their clothing, but no other property; rations from their own stores sufficient to last them beyond our lines; the necessary cooking utensils for preparing their food; and thirty wagons to transport such articles as could not well be carried. These terms I regarded more favorable to the government than an unconditional surrender. It saved us the transportation of them north, which at that time would have been very difficult, owing to the limited amount of river transportation on hand, and the expenses of subsisting them. It left our army free to operate against Johnston, who was threatening us from the direction of Jackson, and our river transportation to be used for the movement of troops to any point the exigency of the service might require.

I deem it proper to state here, in order that the correspondence may be fully understood, that after my answer to General Pemberton's letter of the morning of the 3d, we had a personal interview on the subject of the capitulation.

The particulars and incidents of the siege will be contained in the reports of division and corps commanders, which will be forwarded as soon as received.

I brought forward during the siege, in addition to Lauman's division and four regiments previously ordered from Memphis, Smith's and Kimball's divisions of the sixteenth army corps, and assigned Major-General C. C. Washburne to command of same. On the 11th of June, Major-General F. J. Herron's division from the Department of the Missouri arrived; and on the 14th two divisions of the ninth army corps, Major-General J. G. Parke commanding, arrived. This increase in my force enabled me to make the investment most complete, and at the same time left me a large reserve to watch the movements of Johnston. Herron's division was put into position on the extreme south of the city, and Lauman's division was placed between Herron and McClernand. Smith's and Kimball's divisions and Parke's corps were sent to Haines's Bluff. This place I had fortified to the land side and every preparation made to resist a heavy force. Johnston crossed Big Black river with a portion of his force, and everything indicated that he would make an attack about the 25th of June. Our position in front of Vicksburg having been made as strong against a sortie from the enemy as his works were against an assault, I placed Major-General Sherman in command of all the troops designated to look after Johnston. The force intended to operate against Johnston, in addition to that at Haines's Bluff, was one division from each of the thirteenth, fifteenth, and seventeenth army corps and Lauman's division. Johnston, however, not attacking, I determined to attack him the moment Vicksburg was in our possession, and accordingly notified Sherman that I should again make an assault on Vicksburg at daylight on the 6th, and for him to have up supplies of all descriptions ready to move upon receipt of orders if the assault should prove a success. His preparations were immediately made, and when the place surrendered on the 4th two days earlier than I fixed for the attack, Sherman was found ready, and moved at once with a force increased by the remainder of both the thirteenth and fifteenth army corps, and is at present investing Jackson where Johnston has made a stand.

In the march from Bruinsburg to Vicksburg, covering a period of twenty days, before supplies could be obtained from government stores, only five days' rations were issued, and three days' of these were taken in haversacks at the start, and were soon exhausted. All other subsistence was obtained from the country through which we passed. The march was commenced without wagons, except such as could be picked up through the country. The country was abundantly supplied with corn, bacon, beef and mutton. The troops enjoyed excellent health, and no army ever appeared in better spirits or felt more confident of success.

In accordance with previous instructions, Major-General S. A. Hurlbut started Colonel (now Brigadier-General) B. H. Grierson with a cavalry force from La Grange, Tennessee, to make a raid through the central portion of the State of Mississippi to destroy railroads and other public property, for the purpose of creating a diversion in favor of the army moving to the attack on Vicksburg. On the 17th of April this expedition started, and arrived at Baton Rouge on the 2d of May, having successfully traversed the whole State of Mississippi. This expedition was skillfully conducted and reflects great credit on Colonel Grierson and all of his command. The notice given the raid by the Southern press confirms our estimate of its importance. It has been one of the most brilliant cavalry exploits of the war, and will be handed down in history as an example to be imitated. Colonel Grierson's report is herewith transmitted.

I cannot close this report without an expression of thankfulness for my good fortune in being placed in co-operation with an officer of the navy who accords to every move that seems for the interest and success of our arms his hearty and energetic support. Admiral Porter and the very efficient officers under him have ever shown the greatest readiness in their co-operation, no matter what was to be done or what risk to be taken, either by their men or their vessels. Without this prompt and cordial support my movements would have been much embarrassed, if not wholly defeated.

Captain J. U. Shirk, commanding the Tuscumbia, was especially active and deserving of the highest commendation for his personal attention to the repairing of the damage done our transports by the Vicksburg batteries.

The result of this campaign has been the defeat of the enemy in five battles outside of Vicksburg; the occupation of Jackson, the capital of the State of Mississippi, and the capture of Vicksburg and its garrison and munitions of war; a loss to the enemy of thirty-seven thousand (37,000) prisoners, among whom were fifteen general officers; at least ten thousand men killed and wounded, and among the killed Generals Tracy, Tilghman and Green, and hundreds and perhaps thousands of stragglers, who can never be collected and reorganized. Arms and munitions of war for an army of sixty thousand men have fallen into our hands, besides a large amount of other public property, consisting of railroads, locomotives, cars, steamboats, cotton, &c., and much was destroyed to prevent our capturing it.

Our loss in the series of battles may be summed up as follows: