We have computed the light quark masses using the O(a(2)) improved Alpha ac
tion, in the quenched approximation. The renormalized masses have been obta
ined non-perturbatively. By eliminating the systematic error coming from th
e truncation of the perturbative series, our procedure removes the discrepa
ncies, observed in previous calculations, between the results obtained usin
g the vector and the axial-vector Ward identities. It also gives values of
the quark masses larger than those obtained by computing the renormalizatio
n constants using (boosted) perturbation theory. Our main results, in the R
I (MOM) scheme and at a renormalization scale mu = 2 GeV, are m(s)(RI) = 13
8(15) MeV and m(i)(RI) = 5.6(5) MeV, where m(s)(RI) is the mass of the stra
nge quark and m(l)(RI) = (m(u)(RI) + m(d)(RI))/2 the average mass of the up
-down quarks. From these results, which have been obtained non-perturbative
ly, by using continuum perturbation theory we derive the MS masses, at the
same scale, and the renormalization group invariant (m(RGI)) masses. We fin
d m(s)(NLO MS) = 121(13) MeV and m(l)(NLO MS) = 4.9(4) MeV at the next-to-l
eading order; m(s)(N2 LO MS) = 111(12) MeV, m(l)(N2) (LO) (MS) = 4.5(4) MeV
, m(s)(RGI) = 177(19) MeV and m(l)(RGI) = 7.2(6) MeV at the next-to-next-to
-leading order. (C) 1998 Published by Elsevier Science B.V. All rights rese
rved.