  (abridged) In this work we present the results of one year of upgraded Giant
Metrewave Radio Telescope timing measurements of PSR~J0514$-$4002A, a 4.99-ms
pulsar in a 18.8-day, eccentric ($e \, =\, 0.89$) orbit with a massive
companion located in the globular cluster NGC~1851. Combining these data with
earlier Green Bank Telescope data, we greatly improve the precision of the rate
of advance of periastron, $\dot{\omega} \, = \, 0.0129592(16)\, \deg \, \rm
yr^{-1}$ which, assuming the validity of general relativity, results in a much
refined measurement of the total mass of the binary, $M_{\rm tot} \, = \,
2.4730(6) \, M_{\odot}$. Additionally, we measure the Einstein delay parameter,
$\gamma\,, = \, 0.0216(9) \, \rm s$. Furthermore, we measure the proper motion
of the system ($\mu_{\alpha} \, = \, 5.19(22)$ and $\mu_{\delta} =
-0.56(25)\rm~mas ~ yr^{-1}$), which is not only important for analyzing its
motion in the cluster, but is also essential for a proper interpretation of
$\gamma$, given the latter parameter's correlation with the variation of the
projected semi-major axis. The measurements of $\gamma$ and the proper motion
enable a separation of the system component masses: we obtain a pulsar mass of
$M_{p} \, = \, 1.25^{+0.05}_{-0.06} \, M_{\odot}$ and a companion mass of
$M_{c} \, = \, 1.22^{+0.06}_{-0.05} \, M_{\odot}$. This raises the possibility
that the companion is also a neutron star. Searches for radio pulsations from
the companion have thus far been unsuccessful, hence we cannot confirm the
latter hypothesis. The low mass of this millisecond pulsar - one of the lowest
ever measured for such objects - clearly indicates that the recycling process
can be achieved with a relatively small amount of mass transfer.
