  Weak gravitational lensing measurements based on photometry are limited by
shape noise, the variance in the unknown unlensed orientations of the source
galaxies. If the source is a disk galaxy with a well-ordered velocity field,
however, velocity field data can support simultaneous inference of the shear,
inclination, and position angle, virtually eliminating shape noise. We use the
Fisher Information Matrix formalism to forecast the precision of this method in
the idealized case of a perfectly ordered velocity field defined on an
infinitesimally thin disk. For nearly face-on targets one shear component,
$\gamma_\times$, can be constrained to $0.003\frac{90}{I_0}\frac{25}{n_{\rm
pix}}$ where $I_0$ is the S/N of the central intensity pixel and $n_{\rm pix}$
is the number of pixels across a diameter enclosing 80\% of the light. This
precision degrades with inclination angle, by a factor of three by
$i{=}50^\circ$. Uncertainty on the other shear component, $\gamma_+$, is about
1.5 (7) times larger than the $\gamma_\times$ uncertainty for targets at
$i=10^\circ$ ($50^\circ$). For arbitrary galaxy position angle on the sky,
these forecasts apply not to $\gamma_+$ and $\gamma_\times$ as defined on the
sky, but to two eigenvectors in $(\gamma_+, \gamma_\times,\mu)$ space where
$\mu$ is the magnification. We also forecast the potential of less expensive
partial observations of the velocity field such as slit spectroscopy. We
conclude by outlining some ways in which real galaxies depart from our
idealized model and thus create random or systematic uncertainties not captured
here. In particular, our forecast $\gamma_\times$ precision is currently
limited only by the data quality rather than scatter in galaxy properties
because the relevant type of scatter has yet to be measured.
