  We describe a practical measurement of the curvature of the Universe which
relies purely on the properties of the Robertson-Walker metric rather than any
model for the dynamics or content of the Universe. The cross-correlation
between foreground mass and gravitational shear of background galaxies depends
upon the angular diameter distances d_A(z_l), d_A(z_s), and d_A(z_s,z_l) on the
degenerate triangle formed by observer, source, and lens. In a flat Universe,
d_A(z_l,z_s) = d_A(z_s)-d_A(z_l), but in curved Universes an additional term
$\propto\Omega_k$ alters the lensing observables even if d_A(z) is fixed. Weak
lensing data may be used to solve simultaneously for d_A and the curvature.
This method is completely insensitive to: the equation of state; amendments to
the General Relativity formulae for the deflection of light or the growth of
structure; or biases in the photometric redshift scale. There is however a
degeneracy among d_A, $\Omega_k$ and the galaxy bias factors, that can be
broken by using measurements of baryon acoustic oscillations with the same
imaging data. Ambitious weak-lensing + baryon-oscillation surveys would measure
$\Omega_k$ to an accuracy $\approx0.04 f_{\rm sky}^{-1/2} (\sigma_{\ln
z}/0.04)^{1/2}$, where $\sigma_{\ln z}$ is the photometric redshift error. We
also predict bounds on curvature and other parameters in the context of
specific dark-energy models, and compare to other analyses of the weak lensing
cross-correlation method. We find both curvature and parametric constraints to
be surprisingly insensitive to systematic shear calibration errors.
