We report the discovery of PSR J1757$-$1854, a 21.5-ms pulsar in a highly-eccentric, 4.4-h orbit around a neutron star (NS) companion. PSR J1757$-$1854 exhibits some of the most extreme relativistic parameters of any known pulsar, including the strongest relativistic effects due to gravitational-wave (GW) damping, with a merger time of 76 Myr. Following a 1.6-yr timing campaign, we have measured five post-Keplerian (PK) parameters, yielding the two component masses ($m_text{p}=1.3384(9),text{M}_odot$ and $m_text{c}=1.3946(9),text{M}_odot$) plus three tests of general relativity (GR), which the theory passes. The larger mass of the NS companion provides important clues regarding the binary formation of PSR J1757$-$1854. With simulations suggesting 3-$sigma$ measurements of both the contribution of Lense-Thirring precession to the rate of change of the semi-major axis and the relativistic deformation of the orbit within $sim7-9$ years, PSR J1757$-$1854 stands out as a unique laboratory for new tests of gravitational theories.
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