We have detected low-amplitude radial-velocity variations in two stars, USNO-B1.0 1219-0005465 (GSC 02265-00107 = WASP-1) and USNO-B1.0 0964-0543604 (GSC 00522-01199 = WASP-2). Both stars were identified as being likely host stars of transiting exoplanets in the 2004 SuperWASP wide-field transit survey. Using the newly commissioned radial-velocity spectrograph SOPHIE at the Observatoire de Haute-Provence, we found that both objects exhibit reflex orbital radial-velocity variations with amplitudes characteristic of planetary-mass companions and in-phase with the photometric orbits. Line-bisector studies rule out faint blended binaries as the cause of either the radial-velocity variations or the transits. We perform preliminary spectral analyses of the host stars, which together with their radial-velocity variations and fits to the transit light curves yield estimates of the planetary masses and radii. WASP-1b and WASP-2b have orbital periods of 2.52 and 2.15 d, respectively. Given mass estimates for their F7V and K1V primaries, we derive planet masses 0.80-0.98 and 0.81-0.95 times that of Jupiter, respectively. WASP-1b appears to have an inflated radius of at least 1.33 RJup, whereas WASP-2b has a radius in the range 0.65-1.26 RJup