  We report the first direct evidence for the axisymmetric standard
magnetorotational instability (SMRI) from a combined experimental and numerical
study of a magnetized liquid-metal shear flow in a Taylor-Couette cell with
independently rotating and electrically conducting end caps. When a uniform
vertical magnetic field $B_i$ is applied along the rotation axis, the measured
radial magnetic field $B_r$ on the inner cylinder increases linearly with a
small magnetic Reynolds number $Rm$ due to the magnetization of the residue
Ekman circulation. Onset of the axisymmetric SMRI is identified from the
nonlinear increase of $B_r$ beyond a critical $Rm$ in both experiments and
nonlinear numerical simulations. The axisymmetric SMRI exists only at
sufficiently large $Rm$ and intermediate $B_i$, a feature consistent with
theoretical predictions. Our simulations further show that the axisymmetric
SMRI causes the velocity and magnetic fields to contribute an outward flux of
axial angular momentum in the bulk region, just as it should in accretion
disks.
