We have measured cross sections for the ionization of the K shell by electrons with energies from the respective thresholds up to 100 keV, for Au and Bi. The experimental values are obtained by dividing the number of counts in the Kα peak by the number of counts in an energy interval near the tip of the bremsstrahlung continuum, and multiplying this ratio by the theoretical estimate of bremsstrahlung emission towards the detector in this energy interval. Although such a procedure has already been described in the literature, here it is implemented avoiding some of the simplifications made in earlier works. Our experimental cross sections, which are the first ones to be reported for atoms with > Z 47 close to the threshold, are in reasonable agreement with the theoretical predictions of the semirelativistic distorted-wave Born approximation. Hipplerʼs plane-wave Born approximation with corrections for Coulomb and exchange effects yields cross sections that are closer to the experimental data than those evaluated from the relativistic binary-encounter-Bethe model.