  The Galactic flux of cosmic-ray (CR) positrons in the GeV to TeV energy range
is very likely due to different Galactic components. One of these is the
inelastic scattering of CR nuclei with the atoms of the interstellar medium.
The precise amount of this component determines the eventual contribution from
other sources. We present here a new estimation of the secondary CR positron
flux by incorporating the latest results for the production cross sections of
$e^\pm$ from hadronic scatterings calibrated on collider data. All the
reactions for CR nuclei up to silicon scattering on both hydrogen and helium
are included. The propagation models are derived consistently by fits on
primary and secondary CR nuclei data. Models with a small halo size ($L \leq 2$
kpc) are disfavored by the nuclei data although the current uncertainties on
the beryllium nuclear cross sections may impact this result. The resulting
positron flux shows a strong dependence on the Galactic halo size, increasing
up to factor 1.5 moving $L$ from 8 to 2 kpc. Within the most reliable
propagation models, the positron flux matches the data for energies below 1
GeV. We verify that secondary positrons contribute less than $70\%$ of the data
above a few GeV, corroborating that an excess of positrons is already present
at very low energies. At larger energies, our predictions are below the data
with the discrepancy becoming more and more pronounced. Our results are
provided together with uncertainties due to propagation and hadronic cross
sections. The former uncertainties are below 5\% at fixed $L$, while the latter
are about 7\% almost independently of the propagation scheme. In addition to
the predictions of positrons, we provide new predictions also for the secondary
CR electron flux.
