Well it depends on what you mean by “regiments.” If “regiments” refers to soldiers in the regular army, then no, but there was at least one all-black militia in Louisiana called the Native Guards.

From an article by black historian Henry Louis Gates:

Perhaps the most insidious or desperate attempt to defend the right of black people to own slaves was the statement made on the eve of the Civil War by a group of free people of color in New Orleans, offering their services to the Confederacy, in part because they were fearful for their own enslavement: “The free colored population [native] of Louisiana … own slaves, and they are dearly attached to their native land … and they are ready to shed their blood for her defense. They have no sympathy for abolitionism; no love for the North, but they have plenty for Louisiana … They will fight for her in 1861 as they fought [to defend New Orleans from the British] in 1814–1815.”

These guys were, to put it bluntly, opportunists par excellence: As Noah Andre Trudeau and James G. Hollandsworth Jr. explain, once the war broke out, some of these same black men formed 14 companies of a militia composed of 440 men and were organized by the governor in May 1861 into “the Native Guards, Louisiana,” swearing to fight to defend the Confederacy. Although given no combat role, the Guards — reaching a peak of 1,000 volunteers — became the first Civil War unit to appoint black officers.