Lattice gauge theory techniques have recently achieved sufficient accuracy to permit a determination of the strong coupling constant from the 1P-1S splitting in the charmonium system, with all systematic errors estimated quantitatively. The present result is alpha(MSBAR)(5 GeV) = 0.174 +/- 0.012, or, equivalently, LAMBDA(MS)/(4) = 160(-37)+47 MeV (MSBAR denotes the modified minimal subtraction scheme).