  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.
