We report the first measurement of the elliptic anisotropy (v(2)) of the charm meson D-0 at midrapidity (broken vertical bar y broken vertical bar < 1) in Au + Au collisions at root S-NN = 200 GeV. The measurement was conducted by the STAR experiment at RHIC utilizing a new high- resolution silicon tracker. The measured D-0 v(2) in 0%-80% centrality Au + Au collisions can be described by a viscous hydrodynamic calculation for a transverse momentum (p(T)) of less than 4 GeV/c. The D-0 v(2) as a function of transverse kinetic energy (m(T) - m(0), where m(T) = root p(T)(2) + m(0)(2) p) is consistent with that of light mesons in 10%- 40% centrality Au + Au collisions. These results suggest that charm quarks have achieved local thermal equilibrium with the medium created in such collisions. Several theoretical models, with the temperature-dependent, dimensionless charm spatial diffusion coefficient (2 pi TDs) in the range of similar to 2-12, are able to simultaneously reproduce our D-0 v(2) result and our previously published results for the D-0 nuclear modification factor.