The Elliott-Yafet mechanism is arguably the most promising candidate to explain the ultrafast demagnetization dynamics in ferromagnetic transition metals on timescales on the order of 100 femtoseconds. So far, only electron-phonon scattering has been analyzed as the scattering process needed to account for the demagnetization due to the Elliott-Yafet mechanism. We show for the first time that the electron-electron scattering contribution to the Elliott-Yafet mechanism has the potential to explain time-resolved magneto-optical Kerr effect measurements on thin magnetic cobalt and nickel films, without reference to a phononic "spin bath."