  We consider a relativistic quantum model of confined massive spinning quarks
and antiquarks which describes leading Regge trajectories of mesons. The quarks
are described by the Dirac equations and the gluon contribution is approximated
by the Nambu-Goto straight-line string. The string tension and the current
quark masses are the main parameters of the model. Additional parameters are
phenomenological constants which approximate nonstring short-range
contributions. Comparison of the measured meson masses with the model
predictions allows one to determine the current quark masses (in MeV) to be
$m_s = 227 \pm 5,~ m_c = 1440 \pm 10,~ m_b = 4715 \pm 20$. The chiral $SU_3$
model[23] makes it possible to estimate from here the $u$- and $d$-quark masses
to be $m_u = 6.2 \pm 0.2$~ Mev and $m_d = 11.1 \pm 0.4$ Mev.
