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 { \langle \sigma v \rangle ^ { 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 .