GENERAL—Your note of this date, just received, proposes an armistice of several hours, for the purpose of arranging terms of capitulation through commissioners to be appointed, &c. The effusion of blood you propose stopping by this course, can be ended at any time you may choose, by an unconditional surrender of the city and garrison. Men who have shown so much endurance and courage as those now in Vicksburg, will always challenge the respect of an adversary, and I can assure you will be treated with all the respect due them as prisoners of war. I do not favor the proposition of appointing commissioners to arrange terms of capitulation, because I have no other terms than those indicated above. I am, General, very respectfully, your obedient servant, U. S. GRANT, Major-General. General Bowen, the bearer of General Pemberton’s letter, was received by General A. J. Smith. He expressed a strong desire to converse with General Grant, and accordingly, while declining this, General Grant requested General Smith to say that if General Pemberton desired to see him, an interview would be granted between the lines in McPherson’s front, at any hour in the afternoon which General Pemberton might appoint. A message was soon sent back to General Smith, appointing three o’clock as the hour. General Grant was there with his staff, and with Generals Ord, McPherson, Logan, and A. J. Smith. General Pemberton came late, attended by General Bowen, and Colonel Montgomery. He was much excited and pert in his answers to General Grant. The conversation was held apart between General Pemberton and his officers, and Generals Grant, McPherson, and A. J. Smith. The rebels insisted on being paroled, and allowed to march beyond our lines, officers and men, all with eight days’ rations, drawn from their own stores, the officers to retain their private property and body servants. General Grant heard what they had to say, and left them at the end of an hour and a half, saying that he would send in his ultimatum in writing, to which General Pemberton promised to reply before night, hostilities to cease in the mean time. General Grant then conferred at his headquarters with his corps and division commanders, and sent the following letter to General Pemberton, by the hands of General Logan and Lieutenant-Colonel Wilson. GENERAL GRANT’S OFFER FOR THE SURRENDER. HEADQUARTERS, DEPARTMENT OF TENNESSEE, } NEAR VICKSBURG, July 3, 1863. } Lieutenant-General J. C. PEMBERTON, commanding Confederate forces, Vicksburg, Miss.: