ABSTRACT More than 30 Galactic double neutron star (DNS) binaries have now been identified through radio pulsar timing. The 24 DNSs in the Galactic field with measured total masses lie in the narrow range of 2.3–2.9 M$_{\odot }$. In contrast, gravitational wave (GW) observations have detected two DNS mergers: GW170817, with a total mass of 2.7 M$_{\odot }$, and GW190425, with a significantly higher mass of 3.4 M$_{\odot }$. The unusually high mass of GW190425 suggests a non-standard formation channel not represented in the known Galactic population. To investigate the origin of such a massive DNS system, we model the late evolutionary stages of helium stars with initial masses between 2.5 and 9.8 M$_{\odot }$ in binaries with 1.4 M$_{\odot }$ neutron star companions, using the 1D stellar evolution code mesa at solar metallicity. We test alternative formation pathways and calibrate our models to reproduce the observed Galactic DNS mass and orbital distributions. By incorporating a modified natal kick prescription, our population synthesis results are broadly consistent with the observed total mass distribution of known DNS systems. Only a small fraction of DNSs of our model have total masses ${\ge} 3$ M$_{\odot }$, insufficient to explain the high rate of massive DNS mergers inferred from GW observations. However, our model rules out the formation of heavy DNS systems like GW190425 via the second unstable mass transfer.