  Evidence for static alternating magnetic fields in the vortex cores of
underdoped YBa$_2$Cu$_3$O$_{6+x}$ is reported. Muon spin rotation measurements
of the internal magnetic field distribution of the vortex state of
YBa$_2$Cu$_3$O$_{6.50}$ in applied fields of $H = 1$ T and $H = 4$ T reveal a
feature in the high-field tail of the field distribution which is not present
in optimally doped YBa$_2$Cu$_3$O$_{6.95}$ and which fits well to a model with
static magnetic fields in the vortex cores. The magnitude of the fields is
estimated to be 18(2) G and decreases above $T = 10$ K. We discuss possible
origins of the additional vortex core magnetism within the context of existing
theories.
