We report the first experimental implementation of a method based on simultaneous use of an energy chirp in the electron beam and a tapered undulator, for the generation of ultrashort pulses in a self-amplified spontaneous emission mode free-electron laser (SASE FEL). The experiment, performed at the SPARC FEL test facility, demonstrates the possibility of compensating the nominally detrimental effect of the chirp by a proper taper of the undulator gaps. An increase of more than 1 order of magnitude in the pulse energy is observed in comparison to the untapered case, accompanied by FEL spectra where the typical SASE spiking is suppressed.