Senate resolution declaring the seats of seceding senators to be vacant, March 14, 1861
After Senator Jefferson Davis of Mississippi and other senators representing South Carolina, Alabama, Florida, Georgia, Louisiana, and Mississippi announced their withdrawal, the Senate debated how to define their absence without officially acknowledging the act of secession. In March 1861 the Senate declared those seats vacant. Later, the Senate formally expelled other members who withdrew.
Records of the U.S. Senate, National Archives and Records Administration
Whereas the seats... as members of the Senate have become vacant, therefore resolved that the Secretary be directed to omit their names respectively from the roll
Congress, Secession, and the Confederacy
With the secession of eleven Southern states in 1860 and 1861, the House and Senate lost more than 85 members. Congress quickly acted to ensure that it could continue to fulfill its legislative responsibilities to the Union. One of the first issues members addressed was whether to recognize secession and withdrawal from Congress as legitimate under the Constitution, or simply to declare the Southern congressional seats vacant. Meanwhile, Southerners who departed drew on their experience in Congress to establish a government for the Confederacy.