A new method to determine the peak intensity of focused relativistic laser pulses is experimentally justified. It is based on the measurement of spectra of electrons, accelerated in the beam waist. The detected electrons were emitted from the plasma, generated by nonlinear ionization of low-density gases (helium, argon, and krypton) in the focal area of a laser beam with the peak intensity >1020 W/cm2. The measurements revealed generation of particles with the maximum energy of a few MeV, observed at a small angle relative to the beam axis. The results are supported by numerical particle-in-cell simulations of a laser–low-density plasma interaction. The peak intensity in the focal region derived from experimental data reaches the value of 2.5 × 1020 W/cm2.