GENERAL—In conformity with the agreement of this afternoon, I will submit the following proposition for the surrender of the city of Vicksburg, public stores, &c. On your accepting the terms proposed, I will march in one division, as a guard, and take possession at eight o’clock to-morrow morning. As soon as paroles can be made out and signed by the officers and men, you will be allowed to march out of our lines, the officers taking with them their regimental clothing, and staff, field, and cavalry officers, one horse each. The rank and file will be allowed all their clothing, but no other property. If these conditions are accepted, any amount of rations you may deem necessary can be taken from the stores you now have, and also the necessary cooking utensils for preparing them; thirty wagons also, counting two two-horse or mule teams as one. You will be allowed to transport such articles as cannot be carried along. The same conditions will be allowed to all sick and wounded officers and privates, as fast as they become able to travel. The paroles for these latter must be signed, however, whilst officers are present authorized to sign the roll of prisoners. I am, General, very respectfully, your obedient servant, U. S. GRANT, Major-General. The officer who received this letter, stated that it would be impossible to answer it by night, and it was not till a little before peep of day, that the proposed reply was furnished. While these deliberations were pending, the men of both armies, who simply knew that a surrender was in contemplation, under intense excitement, were anxiously awaiting the result. Groups of soldiers, who a few hours before were engaged in a deathly struggle, now freely engaged in conversation from the edge of the opposing works. REPLY OF GENERAL PEMBERTON, ACCEPTING ALL THE TERMS OFFERED BY GENERAL GRANT. HEADQUARTERS, VICKSBURG, July 3, 1863. Major-General GRANT, commanding United States forces:— GENERAL—I have the honor to acknowledge the receipt of your communication of this date, proposing terms for the surrender of this garrison and post. In the main your terms are accepted; but in justice both to the honor and spirit of my troops, manifested in the defence of Vicksburg, I have the honor to submit the following amendments, which if acceded to by you, will perfect the agreement between us:—At ten o’clock to-morrow I propose to evacuate the works in and around Vicksburg, and to surrender the city and garrison under my command, by marching out with my colors and arms, and stacking them in front of my present limits, after which you will take possession; officers to retain their side arms and personal property, and the rights and property of citizens to be respected.