PSR J1913+1102 is a double neutron star system (DNS) discovered in the Pulsar Arecibo L-band Feed Array survey. We have now very precisely measured the rate of advance of periastron for the system and the Einstein delay. From general relativity, this results in precise mass measurements: 1.65 ± 0.05 and 1.24 ± 0.05 M⊙ for the pulsar and neutron-star companion, respectively. This makes PSR J1913+1102 both the most massive double neutron star system known, and the most asymmetric in mass among compact DNS binaries. This asymmetry will allow for stringent limits on the effects of dipolar gravitational-wave radiation, predicted by alternative theories of gravity, as well as insight into heavy-element production from the eventual merger of this system and others like it. Further observations will also tighten constraints on formation and evolution models; this is crucial for understanding the DNS population, for which there are relatively few mass measurements.