ABSTRACT In this work, we present the results of 1 yr of upgraded Giant Metrewave Radio Telescope timing measurements of PSR J0514−4002A, a 4.99-ms pulsar in a 18.8-d 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, $M_{\rm tot} = 2.4730(6) \, \mathrm{M}_{\odot }$. Additionally, we measure the Einstein delay parameter, γ, something that has never been done for any binary system with an orbital period larger than ${\sim}$10 h. The measured value, $\gamma = 0.0216(9) \, \rm s$, is by far the largest for any binary pulsar. 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 analysing its motion in the cluster, but is also essential for a proper interpretation of γ, given the latter parameter’s correlation with the variation of the projected semimajor axis. The measurements of γ and the proper motion enable a separation of the system component masses: we obtain a pulsar mass of $M_{\rm p} = 1.25^{+0.05}_{-0.06} \, \mathrm{M}_{\odot }$ and a companion mass of $M_{\rm c} = 1.22^{+0.06}_{-0.05} \, \mathrm{M}_{\odot }$. 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.