We report on the detection of GJ3634b, a super-Earth of mass msini=7.0+/-0.9 Mearth and period P=2.64561+/-0.00066-day. Its host star is a M2.5 dwarf, has a mass of 0.45+/-0.05M_{sun}, a radius of 0.43+/-0.03R{sun}_ and lies 19.8+/-0.6pc away from our Sun. The planet is detected after a radial-velocity campaign using the ESO/Harps spectrograph. GJ3634b had an a priori geometric probability to undergo transit of ~7% and, if telluric in composition, a non-grazing transit would produce a photometric dip of <~0.1%. We therefore followed-up upon the RV detection with photometric observations using the 4.5-um band of the IRAC imager onboard Spitzer. Our six-hour long light curve excludes that a transit occurs for 2sigma of the probable transit window, decreasing the probability that GJ3634b undergoes transit to ~0.5%.