We report on International Gamma-Ray Astrophysics Laboratory (INTEGRAL) observations of the soft gamma-ray repeater SGR 1935+2154 performed between 2020 April 28 and May 3. Several short bursts with fluence of similar to 10(-7)-10(-6) erg cm(-2) were detected by the Imager on-board INTEGRAL (IBIS) instrument in the 20-200 keV range. The burst with the hardest spectrum, discovered and localized in real time by the INTEGRAL Burst Alert System, was spatially and temporally coincident with a short and very bright radio burst detected by the Canadian Hydrogen Intensity Mapping Experiment (CHIME) and Survey for Transient Astronomical Radio Emission 2 (STARE2) radio telescopes at 400-800 MHz and 1.4 GHz, respectively. Its lightcurve shows three narrow peaks separated by similar to 29 ms time intervals, superimposed on a broad pulse lasting similar to 0.6 s. The brightest peak had a delay of 6.5 +/- 1.0 ms with respect to the 1.4 GHz radio pulse (that coincides with the second and brightest component seen at lower frequencies). The burst spectrum, an exponentially cutoff power law with photon index Gamma = 0.7(-0.2)(+0.4) and peak energy E-p = 65 +/- 5 keV, is harder than those of the bursts usually observed from this and other magnetars. By the analysis of an expanding dust-scattering ring seen in X-rays with the Neil Gehrels Swift Observatory X-ray Telescope (XRT) instrument, we derived a distance of 4.4(-1.3)(+2.8) kpc for SGR 1935+2154, independent of its possible association with the supernova remnant G57.2+0.8. At this distance, the burst 20-200 keV fluence of (6.1 +/- 0.3) x 10(-7) erg cm(-2) corresponds to an isotropic emitted energy of similar to 1.4 x 10(39) erg. This is the first burst with a radio counterpart observed from a soft gamma-ray repeater and it strongly supports models based on magnetars that have been proposed for extragalactic fast radio bursts.