  The first realization of instabilities in the shear flow between two
superfluids is examined. The interface separating the A and B phases of
superfluid He-3 is magnetically stabilized. With uniform rotation we create a
state with discontinuous tangential velocities at the interface, supported by
the difference in quantized vorticity in the two phases. This state remains
stable and nondissipative to high relative velocities, but finally undergoes an
instability when an interfacial mode is excited and some vortices cross the
phase boundary. The measured properties of the instability are consistent with
a modified Kelvin-Helmholtz theory.
