  We report on spectroscopy of the white-dwarf companion of the millisecond
radio pulsar PSR J1012+5307. We find strong Balmer absorption lines, as would
be expected for a cool DA white dwarf. The profiles are much narrower than
usual, however, and lines are seen up to H12, indicating that the companion has
a low gravity and hence a low mass. This is consistent with the
expectation---based on evolutionary considerations and on the mass
function---that it is a low-mass white dwarf with a helium core. By comparing
the spectra to model atmospheres, we derive an effective temperature
$T_{\rm{}eff}=8550\pm25\,$K and a surface gravity $\log{}g=6.75\pm0.07$ (cgs
units). Using the Hamada-Salpeter mass-radius relation for helium white dwarfs,
with an approximate correction for finite-temperature effects, we infer a mass
$\mwd=0.16\pm0.02\,\msun$. This is the lowest mass among all spectroscopically
identified white dwarfs. We determine radial velocities from our spectra, and
find a radial-velocity amplitude of $280\pm15\,\kms$. With the pulsar's
radial-velocity amplitude, the mass ratio $\mpsr/\mwd=13.3\pm0.7$. From all
constraints, we find that with 95\% confidence $1.5<\mpsr/\msun<3.2$.
