Observations made in February and April 1989 with the Ginga satellite show that the times of maximum light for the 685 s X-ray intensity variations of 4U 1820 - 30 occur 71 + or - 21 and 94 + or - 21 s earlier than would be expected for the constant-period ephemeris based on previous observations. The times of maximum light observed between 1976 and 1989 are consistent with a constant rate period decrease of 0.074 + or - 0.013 ms/yr. The decrease is found to be significant at the 99.9 percent level. If this reflected a true change in the orbital period, this result would be inconsistent with the standard model in which the 685 s period is that of the orbit of a binary star consisting of a neutron star whose companion is a low-mass (0.07 solar mass) degenerate helium star, and mass transfer is driven by loss of orbital angular momentum through gravitational radiation. The apparent period change may be explained in two ways: either the standard model is incorrect, or 4U 1820 - 30 is being accelerated toward the earth by the gravitational field of another body. Acceleration of the binary by a distant third companion in a hierarchical triple, or by the cluster potential, are both possibilities.