  We infer the mean optical depth of a sample of optically-selected galaxy
clusters from the Dark Energy Survey (DES) via the pairwise kinematic
Sunyaev-Zel'dovich (kSZ) effect. The pairwise kSZ signal between pairs of
clusters drawn from the DES Year-3 cluster catalog is detected at $4.1 \sigma$
in cosmic microwave background (CMB) temperature maps from two years of
observations with the SPT-3G camera on the South Pole Telescope. After cuts,
there are 24,580 clusters in the $\sim 1,400$ deg$^2$ of the southern sky
observed by both experiments. We infer the mean optical depth of the cluster
sample with two techniques. The optical depth inferred from the pairwise kSZ
signal is $\bar{\tau}_e = (2.97 \pm 0.73) \times 10^{-3}$, while that inferred
from the thermal SZ signal is $\bar{\tau}_e = (2.51 \pm 0.55^{\text{stat}} \pm
0.15^{\rm syst}) \times 10^{-3}$. The two measures agree at $0.6 \sigma$. We
perform a suite of systematic checks to test the robustness of the analysis.
