We report the first successful attempt to generate ultrashort (1-10 ps) relativistic electron bunches characterized by a ramped longitudinal current profile that rises linearly from head to tail and then falls sharply to zero. Bunches with this type of longitudinal shape may be applied to plasma-based accelerator schemes as an optimized drive beam, and to free-electron lasers as a means of reducing asymmetry in microbunching due to slippage. The scheme used to generate the ramped bunches employs an anisochronous dogleg beam line with nonlinear correction elements to compress a beam having an initial positive time-energy chirp. The beam current profile is measured using a deflecting mode cavity, and a pseudoreconstruction of the beam's longitudinal phase space distribution is obtained by using this diagnostic with a residual horizontal dispersion after the dogleg.