We investigate whether disk accretion can continue as the accreted angular momentum spins the central star to near breakup. We model a steady-state thin accretion disk around a uniformly-rotating unmagnetized star using a two-dimensional fluid with a polytropic equation of state and alpha-viscosity. We explicitly include gradients in the radial velocity and the pressure and numerically solve for the angular velocity profile. We treat the specific angular momentum, j, added to the star as an eigenvalue of the problem that is determined through the boundary conditions. We find that there is a mapping between j and the stellar rotation rate, OMEGA*, with the following properties. When OMEGA* is somewhat less than the breakup rotation rate of the star, OMEGA(max), we find a class of solutions where the angular velocity of the disk attains a maximum close to the star and then decreases rapidly in a boundary layer to match OMEGA*. For a thin disk with thickness approximately 0.01 times the radius and alpha = 0.1, the radial flow of the accreting material briefly becomes supersonic in the boundary layer before being decelerated in a radial shock. For a thicker disk (thickness approximately 0.1 times radius) with much smaller viscosity (alpha = 0.0001), the flow is subsonic throughout. In either case, j is almost independent of OMEGA* and is approximately equal to the Keplerian specific angular momentum at the stellar surface. This agrees with the standard picture of angular momentum transport in thin disks. However, if OMEGA* is near breakup, then we find a second class of solutions where the disk angular velocity has no maximum at all but increases monotonically all the way down to the stellar surface; the flow remains subsonic for all choices of disk thickness and alpha. For these solutions, j decreases extremely rapidly with increasing OMEGA* and even takes on fairly large negative values. Because of this, the spin-up of an accreting star slows down and eventually stops at a rotation rate near breakup. Beyond this point, the star can continue to accrete any amount of matter without actually breaking up. This result has applications in star formation and in the theory of cataclysmic variables. It also eliminates one of the objections to the accretion-induced collapse scenario for the formation of low-mass binary neutron stars.
