  We study junctions between superconductors mediated by the edge states of a
quantum spin Hall insulator. We show that such junctions exhibit a fractional
Josephson effect, in which the current phase relation has a 4\pi, rather than a
2\pi periodicity. This effect is a consequence of the conservation of fermion
parity - the number of electrons modulo 2 - in a superconducting junction, and
is closely related to the Z_2 topological structure of the quantum spin Hall
insulator. Inelastic processes, which violate the conservation of fermion
parity, lead to telegraph noise in the equilibrium supercurrent. We predict
that the low frequency noise due these processes diverges exponentially with
temperature T as T -> 0. Possible experiments on HgCdTe quantum wells will be
discussed.
