We have measured the ion yields for helium ionized by 120 fs, 614 nm laser pulses for intensities near ${10}^{16}$ W/${\mathrm{cm}}^{2}$. We have found that for these ultrashort pulses the ${\mathrm{He}}^{+2}$ data exhibit a feature which saturates in parallel with the ${\mathrm{He}}^{+}$ signal indicating that the ionization may proceed nonsequentially. We propose a new mechanism, which can exist only in the tunneling regime, for such nonsequential ionization.