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We have observed the persistent but optically unidentified X-ray source X1908+075 with the PCA and HEXTE instruments on RXTE. The binary nature of this source was established by Wen, Remillard, & Bradt (2000) who found a 4.4-day orbital period in results from the RXTE ASM. We report the discovery of 605 s pulsations in the X-ray flux. The Doppler delay curve is measured and provides a mass function of 6.1 Msun which is a lower limit to the mass of the binary companion of the neutron star. The degree of attenuation of the low-energy end of the spectrum is found to be a strong function of orbital phase. A simple model of absorption in a stellar wind from the companion star fits the orbital phase dependence reasonably well and limits the orbital inclination angle to the range 38 to 72 degrees. These measured parameters lead to an orbital separation of 60 to 80 lt-s, a mass for the companion star in the range 9-31 Msun, and an upper limit to the size of the companion of ~22 Rsun. From our analysis we also infer a wind mass loss rate from the companion star of >~ 1.3 x 10^-6 Msun/yr and, when the properties of the companion star and the effects of photoionization are considered, likely >~ 4 x 10^-6 Msun/yr. Such a high rate is inconsistent with the allowed masses and radii that we find for a main sequence or modestly evolved star unless the mass loss rate is enhanced in the binary system relative to that of an isolated star. We discuss the possibility that the companion might be a Wolf-Rayet star that could evolve to become a black hole in 10^4 to 10^5 yr. If so, this would be the first identified progenitor of a neutron star--black hole binary.
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