We report the discovery of PSR J1909-3744, a 2.95 ms pulsar in a nearly circular 1.53 day orbit. Its narrow pulse width of 43 μs allows pulse arrival times to be determined with great accuracy. We have spectroscopically identified the companion as a moderately hot (T~8500 K) white dwarf with strong absorption lines. Radial velocity measurements of the companion will yield the mass ratio of the system. Our timing data suggest the presence of Shapiro delay; we expect that further timing observations, combined with the mass ratio, will allow the first accurate determination of a millisecond pulsar mass. We have measured the timing parallax and proper motion for this pulsar, which indicate a transverse velocity of 140+80-40 km s-1. This pulsar's stunningly narrow pulse profile makes it an excellent candidate for precision timing experiments that attempt to detect low-frequency gravitational waves from coalescing supermassive black hole binaries.