The soft X-ray emission seen from SN 1987A and the apparent deceleration of the radio source expansion suggest that the supernova blast wave has encountered a moderately dense H II region interior to the inner circumstellar ring. We simulate the hydrodynamics of this interaction and calculate the resulting X-ray and ultraviolet emission-line spectrum and light curves. The soft X-ray spectrum is dominated by emission lines of hydrogenic and helium-like C, N, O, and Ne; it is consistent with the ROSAT observations if Fe is depleted on grains. N V lambda lambda 1240 emission should be observable easily with the Hubble Space Telescope. The blast wave should strike the inner circumstellar ring around A.D. 2007.