We present solutions for the structure of the boundary layer where the accretion disk meets the neutron star, which is expected to be the dominant source of high-energy radiation in low-mass X-ray binaries which contain weakly magnetized accreting neutron stars. We find that the main portion of the boundary layer gas is hot (> ~10^8 K), low in density, radially and vertically extended, and optically thick to scattering but optically thin to absorption. It will produce large X-ray luminosity by Comptonization. Energy is transported inward by viscosity, concentrating the energy dissipation in the dense, optically thick zone close to the stellar surface. We explore the dependence of the boundary layer structure on the mass accretion rate, the rotation rate of the star, the alpha viscosity parameter and the viscosity prescription. Radiation pressure is the dominant source of pressure in the boundary layer; the flux is close to the Eddington limiting flux even for luminosities well below (~0.01 times) L(Edd). At luminosities near L(Edd), the boundary layer expands radially, and has a radial extent larger than one stellar radius. Based on the temperatures and optical depths which characterize the boundary layer, we expect that Comptonization will produce a power-law spectrum at low source luminosities. At high luminosities, a Planckian spectrum will be produced in the dense region where most of the energy is released, and modified by Comptonization as the radiation propagates outward.

43 pages, 14 Postscript figures, uses aaspp4.sty, submitted to ApJ