The pulse duration in short-pulse schemes for Self-Amplified Spontaneous Emission Free Electron Lasers (SASE FELs) is limited by the FEL coherence time. A recently proposed concept allows to overcome the coherence time barrier and to obtain much shorter pulses. When the lasing part of an electron bunch is much shorter than the coherence time, one can suppress the radiation in the long main undulator while preserving microbunching within that short lasing slice. Then, a short radiation pulse is produced in a relatively short radiator. A possible suppression method, an excessive reverse undulator taper, is discussed and illustrated numerically in this paper. We also performed the first experimental tests of this method at the soft X-ray FEL user facility FLASH. The measured pulse duration approaches 1 fs (FWHM) at the wavelength of 5 nm.