We study the masses of the light quarks with lattice QCD. Most of the dependence on the lattice spacing a, observed previously with Wilson fermions, is removed by an O(a) corrected action. In the quenched approximation, we obtain a strange quark <(MS)over bar> mass (m) over bar(s)(2 GeV) = 95(16) MeV, and an average of the up and down quark masses (m) over bar(l)(2 GeV) = 3.6(6) MeV. Correcting for quenching, the masses likely are 20% to 40% smaller: 54 < (m) over bar(s)(2 GeV) < 92 MeV and 2.1 < (m) over bar(l)(2 GeV) < 3.5 MeV. We argue that most lattice determinations are consistent with these low values, which are outside the range conventionally given.