More precisely, the “no-cloning theorem” of quantum mechanics implies that one cannot replicate the exact quantum state of a clump of matter in a second clump of matter (although it is possible to “teleport” the exact quantum state by transmitting its information, provided one is willing to mess up the original).
However, although no one knows for sure, it seems likely that the operations of consciousness are purely “classical” in the sense that the exact quantum state is not needed (Roger Penrose’s ideas not withstanding).
Turns out that while the no cloning theorem does prohibit cloning, it does not prohibit perfect replication as long as the original is destroyed. This is the premise of quantum teleportation. Using some clever applications of entanglement it is possible to perfectly replicate the quantum state of a particle in a different location, it's just that the original particle is necessarily destroyed in the process.
So, in short the no cloning theorem does not prevent quantum teleportation.
In this sense, the follow-up in the paradox where you create 10 copies all of which are identical would not work. Quantum mechanics effectively ensures that there's ever only at most one of you.
However, although no one knows for sure, it seems likely that the operations of consciousness are purely “classical” in the sense that the exact quantum state is not needed (Roger Penrose’s ideas not withstanding).