The motion of a spacecraft about a comet is studied under the assumption that the spacecraft is in the vicinity of the body (but far enough so that the higher gravitational moments can be neglected) and is subject to its gravitational attraction modeled as a point mass, the solar gravitational attraction and the solar radiation pressure. The comet is assumed to have an eccentric orbit about the Sun. The equations of motion for this situation are derived applying the Hill approximation. The equilibrium solutions of the problem are defined and their stability properties are studied. A significant result is that the location of the Sun-side equilibrium points may be pushed far from the body and their characteristic instability time increased to the order of years, implying that it may be feasible to station a spacecraft at such a point to monitor a comet during its perihelion passage. Arising from this analysis is also a sufficient condition for the spacecraft orbit to remain bound to the comet. This condition is verified numerically for a limited number of cases. The analysis in this paper also applies to the motion of the spacecraft about an asteroid or to the natural motion of small particles about a comet or asteroid.