We report the discovery and initial follow-up of a double neutron star (DNS) system, PSR J1946$+$2052, with the Arecibo L-Band Feed Array pulsar (PALFA) survey. PSR J1946$+$2052 is a 17-ms pulsar in a 1.88-hour, eccentric ($e \, =\, 0.06$) orbit with a $\gtrsim 1.2 \, M_\odot$ companion. We have used the Jansky Very Large Array to localize PSR J1946$+$2052 to a precision of 0.09 arcseconds using a new phase binning mode. We have searched multiwavelength catalogs for coincident sources but did not find any counterparts. The improved position enabled a measurement of the spin period derivative of the pulsar ($\dot{P} \, = \, 9\,\pm \, 2 \,\times 10^{-19}$); the small inferred magnetic field strength at the surface ($B_S \, = \, 4 \, \times \, 10^9 \, \rm G$) indicates that this pulsar has been recycled. This and the orbital eccentricity lead to the conclusion that PSR J1946$+$2052 is in a DNS system. Among all known radio pulsars in DNS systems, PSR J1946$+$2052 has the shortest orbital period and the shortest estimated merger timescale, 46 Myr; at that time it will display the largest spin effects on gravitational wave waveforms of any such system discovered to date. We have measured the advance of periastron passage for this system, $\dot{\omega} \, = \, 25.6 \, \pm \, 0.3\, \deg \rm yr^{-1}$, implying a total system mass of only 2.50 $\pm$ 0.04 $M_\odot$, so it is among the lowest mass DNS systems. This total mass measurement combined with the minimum companion mass constrains the pulsar mass to $\lesssim 1.3 \, M_\odot$.