We report observations of the remnant of supernova 1987A with the High Resolution Camera (HRC) on board the Chandra X-ray Observatory. A direct image from the HRC resolves the annular structure of the X-ray remnant, confirming the morphology previously inferred by deconvolution of lower resolution data from the Advanced CCD Imaging Spectrometer. Detailed spatial modeling shows that a thin ring plus a thin shell gives statistically the best description of the overall remnant structure, and suggests an outer radius of 096 ± 005 ± 003 for the X-ray-emitting region, with the two uncertainties corresponding to the statistical and systematic errors, respectively. This is very similar to the radius determined by a similar modeling technique for the radio shell at a comparable epoch, in contrast to previous claims that the remnant is 10%-15% smaller at X-rays than in the radio band. The HRC observations put a flux limit of 0.010 counts s–1 (99% confidence level, 0.08-10 keV range) on any compact source at the remnant center. Assuming the same foreground neutral hydrogen column density as toward the remnant, this allows us to rule out an unobscured neutron star with surface temperature T ∞ > 2.5 MK observed at infinity, a bright pulsar wind nebula or a magnetar.