States have no authority to secede; even the Confederacy recognized that. The only real check on federal power is an informed, vigilant electorate of good moral substance holding its elected leaders accountable for their fidelity to the Constitution.
Where does it say - in the Constitution, the state Constitutions, or any law for that matter - that states don't have the right to secede? Hell, two states - Arizona and Texas - have the right to secede written into THEIR Constitutions, and refused to join the US unless that right was confirmed. Furthermore, my reading of the Constitution turns up the 10th Amendment, which says:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
Since the Constitution says nothing about preventing states from seceding, or delegating to the US any power regarding secession either way, it seems to me that that power is, then, reserved to the States and the people (people can, of course, remove themselves from the United States at will).