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 to distinguish two concentrated galaxy nuclei at small separations. This method does not rely on low surface brightness features to identify mergers, and is therefore reliable to high redshift. Using mock images, we derive statistical contamination and incompleteness corrections for the fraction of late-stage mergers. The mock images show that our method returns an uncontaminated (<10%) sample of mergers with projected separations between 2.2 and 8kpc out to z~1. We apply our new method to a magnitude-limited (m_FW814_<23) sample of 44164 galaxies from the COSMOS HST/ACS catalog. Using a mass-complete sample with log M_{star}_/M_{sun}_>10.6 and 0.25<z<=1.00, we find ~5% of systems are late-stage mergers. Correcting for incompleteness and contamination, the fractional merger rate increases strongly with redshift as {\bf R}_merge_\propto(1+z)^3.8+/-0.9^, in agreement both 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 with redshift, (1+z)^4.5+/-1.3^, while the merger rate for quiescent galaxies is consistent with no evolution, (1+z)^1.1+/-1.2^. The merger rate also becomes steeper with decreasing stellar mass. Limiting our sample to galaxies with spectroscopic redshifts from zCOSMOS, we find that the star formation rates and X-ray selected active galactic nucleus (AGN) activity in likely late-stage mergers are higher by factors of ~2 relative to those of a control sample. Combining our sample with more widely separated pairs, we find that 8+/-5% of star formation and 20+/-8% of AGN activity are triggered by close encounters (<143kpc) or mergers, providing additional evidence that major mergers are not the only channels for star formation and black hole growth.