We report the first measurement of the elliptic anisotropy (v_{2}) of the charm meson D^{0} at midrapidity (|y|<1) in Au+Au collisions at sqrt[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}=sqrt[p_{T}^{2}+m_{0}^{2}]) 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 (2πTD_{s}) in the range of ∼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.