A technique for measuring directly the angular distributions, and consequently the absolute differential cross sections, for positron‐ and electron‐atom elastic scattering is described. By exploiting time‐of‐flight spectrometry, simultaneous measurement over a wide range of forward angles is achieved. The structure and performance of the components of the experimental system are described in detail, and the reliability of the apparatus in the measurement of positron‐atom cross sections is tested by using the same spectrometer to study electron scattering. Results are presented for elastic electron‐argon scattering at energies between 5 and 20 eV which are in reasonable agreement with other recent results. Systematic effects on the measurements are discussed.