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A Great Battle on the Rappahannock.

The air has been full of rumors for two or three days past of fighting on the south side of the Rappahannock. Indeed it could not but be assumed that such would be the result of the movements known for some days past to be in progress. Yesterday these rumors began to assume definiteness, no longer leaving any doubt in the public mind that a great battle has taken place. There was an intense eagerness to learn the news on the part of all classes of citizens, which will be rather increased than diminished this morning.

Fighting commenced on Saturday afternoon, between the corps of Gen. Hooker, which had crossed the Rappahannock above Fredericksburg, and a large force of the rebels who attempted to resist the advance of the former to the rear of Fredericksburg. The Union forces, with the exception of a portion of Gen. Howard's corps, behaved nobly. Especial praise is bestowed upon General Sickles and his brave veterans, who seem to have stood the most violent assaults of the rebels with the utmost firmness. A battery lost by the misbehavior of a portion Gen. Howard's Division, was gallantly recaptured by a night attack on Saturday night, and other important advantages gained.

On Sunday the battle was renewed, and raged fiercely throughout the day. Gen. Sedgwick crossed the Rappahannock in the face of the enemy at Fredericksburg, subsequently carrying the heights at the point of the bayonet, and capturing a large number of prisoners. Among the prisoners are said to be the celebrated Washington Artillery of New Orleans, a crack battery organized at the very beginning of the war, and composed of young men from the aristocratic families of that city.

The summing up of the result implies a decided advantage on the Union side, though the final result remains undetermined. Five thousand prisoners are reported captured, among whom are Gens. Evans, of South Carolina, and Fitzhugh Lee, (son of the rebel commander-in-chief.) There is a report, also, that the rebel Gen. Hill is among the killed. We are called to mourn the loss of Maj. Gen. Berry (successor to Gen. Hooker's late command) and a large number of inferior officers, and many of the rank and file. Yet there is reason to believe that the loss of the enemy far outnumbers our own.

It is rendered almost certain that Gen. Stoneman's cavalry has performed noble service in cutting the railroad and destroying the bridges south of Fredericksburg, thus cutting off the rebel retreat and preventing the arrival of reinforcements from Richmond. The reported capture of Gordonsville, though a measure of the utmost importance, and probably a part of the programme, we are inclined to receive with a share of allowance, or at least regard as premature.

There is no longer any room to doubt that the rebel plans have been seriously deranged. The reports which come from Fortress Monroe of the sudden retirement of the rebels from the front of Suffolk gives ground to the belief that they are needed to reinforce the rebel army in front of Hooker. It is almost certain, however, that they will find some difficulty in forming a junction.

Our readers will refer to our dispatches, which are very full, for details. On the whole, the prospect is very cheering for the national cause, and will justify anticipations of the most important success that has occurred during the war.