A dispatch received this morning from Baton Rouge announces the fact that two regiments of Illinois cavalry (the Sixth and Seventh), 900 strong, with a battery of artillery, under command of Colonels Grierson and Prince, left La Grange, near Memphis, Tenn., and arrived on the 2nd instant at Baton Rouge, after a march of seventeen days, with the loss of only 4 men, destroying large quantities of provisions, burning many bridges, cars, and engines on the Jackson road, and capturing and paroling 1,600 prisoners. If the colonel would join us at Alexandria we could capture the Legislature that commences its session this month at Shreveport or compel their adjournment to Texas. I shall invite him to do so, assuring him in that event he may return, by the way of Balck River, through Arkansas, Illinois, and Kentucky, to Tennessee.

My command moved this morning in the direction of Alexandria, General Dwight's brigade, of Gover's division, being in the advance. The other divisions will move forward to-morrow. The purpose movement is to ascertain definitely the strength and purpose of the enemy at Alexandria, and to occupy that place or not, as events shall justify. I will report to you regularly our progress and results. Brigadier-General Emory leaves his division this day, on account of severe illness, and goes to New Orleans for twenty days. His division is place temporarily under the command of Brigadier-General Weitzel, whose brigade in the mean time has been ordered to report to General Dwight, in the advance.

I have the honor to be, with much respect, your obedient servant,

N. P. BANKS,

Major-General, Commanding.

Major-General HALLECK,

Commander-in-Chief of the Army of the United States.

[Inclosure No. 1.]

FLAG-SHIP HARTFORD,

Off Red River, May 1, 1863.

Major-General BANKS:

DEAR GENERAL: Two hours before the arrival of the Arizona I sent the Switzerland up to Grand Gulf to communicate with General Grant and Admiral Porter. They are very busy at work on Vicksburg. He has sent 1,500 cavalry to break up the railroad leading to Jackson. I heard of the gunboats at Grand Gulf, and so sent to beg for two iron-clads to co-operate with you, and hope to have them, but if my gunboats come up I will be with you at Alexandria. They have deserted Fort De Russy and taken the guns to Alexandria, and say they will make their stand there. The guns are all removed from Fort De Russy. We saw a man who saw the guns on the boat taking them up; but there is a general stampede, and we ought to press on. I will send the gunboats to the mouth of Black River, where Kirby Smith is said to be gone up to Arkansas for re-enforcements, and we can capture them if we get there in time. You say truly, general, that everything is dependent upon time; that is, celerity of movement will keep up the panic, although I believe that General Grant is about to attack to city of Vicksburg. Still I hope to get one or two iron-clads. Alexandria is their great machine-shop; break them up and they are more injured than in any other way. If they have time the Webb and other steamers will get above the fails. I was going down to communicate with Alden