We use four deep serendipitous fields observed with the HST Wide-Field Camera to constrain the rate of galaxy merging between the current epoch and z congruent-to 0.7. Since most mergers occur between members of bound pairs, the merger rate is given to a good approximation by (half) the rate of disappearance of galaxies in pairs. An objective criterion for pair membership shows that 34% +/- 9% of our HST galaxies with I = 18-22 belong to pairs, compared to 7% locally. This means that about 13% of the galaxy population has disappeared due to merging in the cosmic epoch corresponding to this magnitude interval (or 0.1 less than or similar to z less than or similar to 0.7). Our pair fraction is a lower limit: correction for pair members falling below our detection threshold might raise the fraction to approximately 50%. Since we address only two-galaxy merging, these values do not include physical systems of higher multiplicity. Incorporating I-band field-galaxy redshift distributions, the pair fraction grows with redshift as is-proportional-to (1 + Z)3.5+/-0.5 and the merger rate as (1 + z)2.5+/-0.5. This may have significant implications for the interpretation of galaxy counts (disappearance of faint blue galaxies), the cosmological evolution of faint radio sources and quasars [which evolve approximately as is-proportional-to (1 + Z)3; the similarity in the power law is necessary but not sufficient evidence for a causal relation], statistics of QSO companions, the galaxy content in distant clusters, and the merging history of a ''typical'' galaxy.