  Clusters of galaxies are expected to gravitationally lens the cosmic
microwave background (CMB) and thereby generate a distinct signal in the CMB on
arcminute scales. Measurements of this effect can be used to constrain the
masses of galaxy clusters with CMB data alone. Here we present a measurement of
lensing of the CMB by galaxy clusters using data from the South Pole Telescope
(SPT). We develop a maximum likelihood approach to extract the CMB cluster
lensing signal and validate the method on mock data. We quantify the effects on
our analysis of several potential sources of systematic error and find that
they generally act to reduce the best-fit cluster mass. It is estimated that
this bias to lower cluster mass is roughly $0.85\sigma$ in units of the
statistical error bar, although this estimate should be viewed as an upper
limit. We apply our maximum likelihood technique to 513 clusters selected via
their SZ signatures in SPT data, and rule out the null hypothesis of no lensing
at $3.1\sigma$. The lensing-derived mass estimate for the full cluster sample
is consistent with that inferred from the SZ flux: $M_{200,\mathrm{lens}} =
0.83_{-0.37}^{+0.38}\, M_{200,\mathrm{SZ}}$ (68% C.L., statistical error only).
