We report the results of 19 years of Arecibo timing for two pulsars in the globular cluster NGC 5904 (M5), PSR B1516+02A (M5A) and PSR B1516+02B (M5B). This has resulted in the measurement of the proper motions of these pulsars and, by extension, that of the cluster itself. M5B is a 7.95 ms pulsar in a binary system with a >0.13 M⊙ companion and an orbital period of 6.86 days. In deep HST images, no optical counterpart is detected within ~2.5 σ of the position of the pulsar, implying that the companion is either a white dwarf or a low-mass main-sequence star. The eccentricity of the orbit (e = 0.14) has allowed a measurement of the rate of advance of periastron: dot omega = 0.0142° +/- 0.0007° yr-1. We argue that it is very likely that this periastron advance is due to the effects of general relativity, the total mass of the binary system then being 2.29 +/- 0.17 M⊙. The small measured mass function implies, in a statistical sense, that a very large fraction of this total mass is contained in the pulsar: Mp = 2.08 +/- 0.19 M⊙ (1 σ) there is a 5% probability that the mass of this object is <1.72 M⊙ and a 0.77% probability that 1.2 M⊙ <= Mp <= 1.44 M⊙. Confirmation of the median mass for this neutron star would exclude most "soft" equations of state for dense neutron matter. Millisecond pulsars (MSPs) appear to have a much wider mass distribution than is found in double neutron star systems; about half of these objects are significantly more massive than 1.44 M⊙. A possible cause is the much longer episode of mass accretion necessary to recycle a MSP, which in some cases corresponds to a much larger mass transfer.