  We argue that the anomalously high fluxes of positrons and antiprotons found
in cosmic rays (CR) can be satisfactorily explained by introducing two
additional elements to the current "standard" paradigm of Galactic CRs. First,
we propose that the antiparticles are effectively produced in interactions of
primary CRs with the surrounding gas not only in the interstellar medium (ISM)
but also inside the accelerators. Secondly, we postulate the existence of two
source populations injecting CRs into the ISM with different, (1) soft (close
to $FI \propto E^{-2.3}$) and (2) hard ($FII \propto E^{-1.8}$ or harder),
energy distributions. Assuming that CRs in the 2nd population of accelerators
accumulate "grammage" of the order of $1 \ \rm g/cm^2$ before their leakage
into ISM, we can explain the energy distributions and absolute fluxes of both
positrons and antiprotons, as well as the fluxes of secondary nuclei of the
(Li,Be,B) group. The superposition of contributions of two source populations
also explains the reported hardening of the spectra of CR protons and nuclei
above 200 GV. The 2nd source population accelerating CRs with a rate at the
level below 10 percent of the power of the 1st source population, can be
responsible for the highest energy protons and nuclei of Galactic CRs up to the
"knee" around $10^{15} \ \rm eV$.
