  Anisotropic lattices, with a temporal lattice spacing smaller than the
spatial one, allow precision Monte Carlo calculations of problems that are
difficult to study otherwise: heavy quarks, glueballs, hybrids, and high
temperature thermodynamics, for example. We here perform the first step
required for such studies with the (quenched) Wilson gauge action, namely, the
determination of the renormalized anisotropy $\xi$ as a function of the bare
anisotropy $\xi_0$ and the coupling. By, essentially, comparing the
finite-volume heavy quark potential where the quarks are separated along a
spatial direction with that where they are separated along the time direction,
we determine the relation between $\xi$ and $\xi_0$ to a fraction of 1% for
weak and to 1% for strong coupling. We present a simple parameterization of
this relation for $1\leq \xi \leq 6$ and $5.5 \leq \beta \leq \infty$, which
incorporates the known one-loop result and reproduces our non-perturbative
determinations within errors. Besides solving the problem of how to choose the
bare anisotropies if one wants to take the continuum limit at fixed
renormalized anisotropy, this parameterization also yields accurate estimates
of the derivative $\partial\xi_0/\partial\xi$ needed in thermodynamic studies.
