  The role of major mergers in galaxy and black hole formation is not well
constrained. To help address this, we develop an automated method to identify
late-stage galaxy mergers before coalescence of the galactic cores. The
resulting sample of mergers is distinct from those obtained using pair-finding
and morphological indicators. Our method relies on median-filtering of
high-resolution images in order to distinguish two concentrated galaxy nuclei
at small separations. Using mock images, we derive statistical contamination
and incompleteness corrections for the fraction of late-stage mergers. We apply
our new method to a magnitude-limited (I < 23) sample of 44,164 galaxies from
the COSMOS HST/ACS catalog. Using a mass-complete sample with $\log M_*/M_\odot
> 10.6$ and $0.25 < z \leq 1.00$, we find ~5% of systems are late-stage mergers
with separations between 2.2 and 8 kpc. Correcting for incompleteness and
contamination, the fractional merger rate increases strongly with redshift as
$(1+z)^{3.8\pm0.9}$, in agreement with earlier studies and with dark matter
halo merger rates. Separating the sample into star-forming and quiescent
galaxies shows that the merger rate for star-forming galaxies increases
strongly redshift, $(1+z)^{4.5\pm1.3}$, while the merger rate for quiescent
galaxies is consistent with no evolution, $(1+z)^{1.1\pm1.2}$. Limiting our
sample to galaxies with spectroscopic redshifts from zCOSMOS, we find that the
star formation rates and X-ray selected AGN activity in likely late-stage
mergers are enhanced by factors of ~2 relative to a control sample. Combining
our sample with more widely separated pairs, we find that $8\pm5\%$ of star
formation and $20\pm8\%$ of AGN activity is triggered by close encounters (<143
kpc) or mergers, once more suggesting that major mergers are not the only
channels for star formation and black hole growth. (abridged)
