The dynamics of water dimers was investigated at the single-molecule level by using a scanning tunneling microscope. The two molecules in a water dimer, bound on a Cu(110) surface at 6 K, were observed to exchange their roles as hydrogen-bond donor and acceptor via hydrogen-bond rearrangement. The interchange rate is approximately 60 times higher for (H2O)2 than for (D2O)2, suggesting that quantum tunneling is involved in the process. The interchange rate is enhanced upon excitation of the intermolecular mode that correlates with the reaction coordinate.