This study is devoted to the theory and calculation of cross sections for the electron-impact excitation of doubly-excited states in helium which are stable to autoionization. The cross sections are found to exhibit sharp peaks just above the threshold energies for excitation and to decrease rapidly with further increase of energy. The maximum value of the Born-Oppenheimer cross section for excitation of the (2p)2 3Pg state is about 6×10-4 (ao)2 and occurs at approximately 11 eV above threshold. The cross sections for excitation of the 2p3p 1Pg, 2p3d 3Du, and 2p3d1Du states are all less than 35 b. The cross section for the (2p)2 3Pg state has been calculated according to the Born-Oppenheimer, distorted-wave, and two-state strong-coupling approximations. In all of these cases the maximum amplitude of the scattered beam occurs at right angles to the direction of the incident electrons. The cross sections of the potential-and-exchange-distortion method (DEW) and of the complete two-state strong-coupling approximation are virtually identical. The Born-Oppenheimer approximation produces remarkably similar results. Rigorous upper bounds to the energies of the 2p3p 1Pg, 2p3d 3Du, and 2p3d 1Du states of helium are reported. Finally, the cross sections are calculated in Born-Oppenheimer approximation for electron-impact excitation of the (1s)2(2p)2 3Pg and (1s)22p3p 3Pg states of beryllium. The peak values of these cross sections are 12 (ao)2 and 2(ao)2, respectively.