  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)$\times10^{-7}$,
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 $0.2 M_\odot$.
