Absolute measurements with an accuracy of 22% and theoretical results in a distorted-wave Born approximation (DWBA) are reported for the triple-differential cross section for 26.6-eV electron-impact ionization of helium. An apparatus is used that allows all scattering angles to be independently varied for both coplanar and noncoplanar geometries. The measurements are compared with a DWBA calculation that includes exchange distortion in the calculation of the distorted waves, as well as with earlier calculations by Crothers [J. Phys. B 19, 463 (1986)] and Pan and Starace [Phys. Rev. Lett. 67, 185 (1991)]. Emphasis is placed on understanding the mechanisms for near-threshold ionization