The VERITAS array of Cherenkov telescopes has carried out a deep observational program on the nearby dwarf spheroidal galaxy Segue 1. We report on the results of nearly 48 hours of good quality selected data, taken between January 2010 and May 2011. No significant $gamma$-ray emission is detected at the nominal position of Segue 1, and upper limits on the integrated flux are derived. According to recent studies, Segue 1 is the most dark matter-dominated dwarf spheroidal galaxy currently known. We derive stringent bounds on various annihilating and decaying dark matter particle models. The upper limits on the velocity-weighted annihilation cross-section are $mathrm{<sigma v >^{95% CL} lesssim 10^{-23} cm^{3} s^{-1}}$, improving our limits from previous observations of dwarf spheroidal galaxies by at least a factor of two for dark matter particle masses $mathrm{m_{chi}gtrsim 300 GeV}$. The lower limits on the decay lifetime are at the level of $mathrm{tau^{95% CL} gtrsim 10^{24} s}$. Finally, we address the interpretation of the cosmic ray lepton anomalies measured by ATIC and PAMELA in terms of dark matter annihilation, and show that the VERITAS observations of Segue 1 disfavor such a scenario.
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