We report on nearly two years of timing observations of the low-mass binary millisecond pulsar , PSR J1909 - 3744 with the Caltech-Parkes-Swinburne Recorder II ( CPSR2 ) , a new instrument that gives unprecedented timing precision . Daily observations give a weighted rms residual of 74 ns , indicating an extremely low level of systematic error . We have greatly improved upon the previous parallax and proper motion measurements of PSR J1909 - 3744 , yielding a distance of ( 1.14 ^ { +0.04 } _ { -0.03 } ) kpc and transverse velocity of ( 200 ^ { +7 } _ { -6 } ) km s ^ { -1 } . The system ’ s orbital eccentricity is just 1.35 ( 12 ) \times 10 ^ { -7 } , the smallest yet recorded . Since their discovery , the masses of the rapidly rotating millisecond pulsars have remained a mystery , with the recycling hypothesis arguing for heavy objects , and the accretion-induced collapse of a white dwarf more consistent with neutron stars less than the Chandrashkar limit . Fortuitously , PSR J1909 - 3744 is an edge-on system , and our data have allowed the measurement of the range and shape of the Shapiro delay to high accuracy , giving the first precise determination of a millisecond pulsar mass to date , m _ { p } = ( 1.438 \pm 0.024 ) M _ { \odot } . The mass of PSR J1909 - 3744 is at the upper edge of the range observed in mildly recycled pulsars in double neutron star systems , consistent with the the recycling hypothesis . It appears that the production of millisecond pulsars is possible with the accretion of less than 0.2 M _ { \odot } .