We have detected quasi-periodic oscillations (QPOs) near 1 kHz from the low mass X-ray binary 4U 0614+091 in observations with XTE. The observations span several months and sample the source over a large range of X-ray luminosity. In every interval QPOs are present above 400 Hz with fractional RMS amplitudes from 3 to 12%. At high count rates, two high frequency QPOs are detected simultaneously. The difference of their frequency centroids is consistent with 323 Hz in all observations. During one interval a third signal is detected at 328 +/- 2 Hz. This suggests the system has a stable `clock which is most likely the neutron star with spin period 3.1 msec. Thus, our observations and those of another neutron star system by Strohmayer et al. (1996) provide the first evidence for millisecond pulsars within low-mass X-ray binary systems and reveal the `missing-link between millisecond radiopulsars and the late stages of binary evolution in low mass X-ray binaries (Alpar et al. 1982). We suggest that the kinematics of the magnetospheric beat-frequency model (Alpar and Shaham 1985) applies to these QPOs. In this interpretation the high frequency signal is associated with the Keplerian frequency of the inner accretion disk and the lower frequency `beat signal arises from the differential rotation frequency of the inner disk and the spinning neutron star. Assuming the high frequency QPO is a Keplerian orbital frequency for the accretion disk, we find a maximum mass of 1.9 solar masses and a maximum radius of 17 km for the neutron star.