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 solar mass companion and an orbital period of 6.86 days. In deep HST images, no optical counterpart is detected within ~2.5 sigma 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: (0.0142 +/-0.0007) degrees per year. 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 solar masses. The small measured mass function implies, in a statistical sense, that a very large fraction of this total mass is contained in the pulsar: 2.08 +/- 0.19 solar masses (1 sigma); there is a 5% probability that the mass of this object is < 1.72 solar masses and a 0.77% probability that is is between 1.2 and 1.44 solar masses. 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 solar masses. 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.

10 pages in ApJ emulate format, 2 tables, 6 figures. Added February 2008 data, slightly revised mass limits. Accepted for publication in ApJ