  We have computed the light quark masses using the O(a^2) improved Alpha
action, in the quenched approximation. The renormalized masses have been
obtained non-perturbatively. By eliminating the systematic error coming from
the truncation of the perturbative series, our procedure removes the
discrepancies, observed in previous calculations, between the results obtained
using the vector and the axial-vector Ward identities. It also gives values of
the quark masses larger than those obtained by computing the renormalization
constants using (boosted) perturbation theory. Our main results, in the RI
(MOM) scheme and at a renormalization scale \mu=2 GeV, are m^{RI}_s= 138(15)
MeV and m^{RI}_l= 5.6(5) MeV, where m^{RI}_s is the mass of the strange quark
and m^{RI}_l=(m^{RI}_u+m^{RI}_d)/2 the average mass of the up-down quarks. From
these results, which have been obtained non-perturbatively, by using continuum
perturbation theory we derive the \bar{MS} masses, at the same scale, and the
renormalization group invariant (m^{RGI}) masses. We find m^{NLO \bar{MS}}_s=
121(13)$ MeV and m^{NLO\bar{MS}}_l= 4.9(4) MeV at the next-to-leading order;
m^{N^2LO \bar{MS}}_s= 111(12) MeV, m^{N^2LO \bar{MS}}_l= 4.5(4) MeV, m_s^{RGI}=
177(19) MeV and m^{RGI}_l= 7.2(6) MeV at the next-to-next-to-leading order.
