  The MMGPS-L is the most sensitive pulsar survey in the Southern Hemisphere.
We present a follow-up study of one of these new discoveries, PSR J1208-5936, a
28.71-ms recycled pulsar in a double neutron star system with an orbital period
of Pb=0.632 days and an eccentricity of e=0.348. Through timing of almost one
year of observations, we detected the relativistic advance of periastron
(0.918(1) deg/yr), resulting in a total system mass of Mt=2.586(5) Mo. We also
achieved low-significance constraints on the amplitude of the Einstein delay
and Shapiro delay, in turn yielding constraints on the pulsar mass
(Mp=1.26(+0.13/-0.25) Mo), the companion mass (Mc=1.32(+0.25/-0.13) Mo, and the
inclination angle (i=57(12) degrees). This system is highly eccentric compared
to other Galactic field double neutron stars with similar periods, possibly
hinting at a larger-than-usual supernova kick during the formation of the
second-born neutron star. The binary will merge within 7.2(2) Gyr due to the
emission of gravitational waves. With the improved sensitivity of the MMGPS-L,
we updated the Milky Way neutron star merger rate to be 25(+19/-9) Myr$^{-1}$
within 90% credible intervals, which is lower than previous studies based on
known Galactic binaries owing to the lack of further detections despite the
highly sensitive nature of the survey. This implies a local cosmic neutron star
merger rate of 293(+222/-103} Gpc/yr, consistent with LIGO and Virgo O3
observations. With this, we predict the observation of 10(+8/-4) neutron star
merger events during the LIGO-Virgo-KAGRA O4 run. We predict the uncertainties
on the component masses and the inclination angle will be reduced to
5x10$^{-3}$ Mo and 0.4 degrees after two decades of timing, and that in at
least a decade from now the detection of the shift in Pb and the sky proper
motion will serve to make an independent constraint of the distance to the
system.
