Context. The precise mechanisms that provide the nonradiative energy for heating the chromosphere and corona of the Sun and other stars are at the focus of intense contemporary research. Aims.- Observations at submm and mm wavelengths are particularly useful to obtain information about the run of the temperature in the upper atmosphere of Sun-like stars. We used the Atacama Large Millimeter/submillimeter Array (ALMA) to study the chromospheric emission of the α Centauri binary system in all six available frequency bands during Cycle 2 in 2014-2015. Methods- Since ALMA is an interferometer, the multitelescope array is particularly suited for the observation of point sources. With its large collecting area, the sensitivity is high enough to allow the observation of nearby main-sequence stars at submm/mm wavelengths for the first time. The comparison of the observed spectral energy distributions with theoretical model computations provides the chromospheric structure in terms of temperature and density above the stellar photosphere and the quantitative understanding of the primary emission processes. Results- Both stars in the α Centauri binary system were detected and resolved at all ALMA frequencies. For both α Cen A and B, the existence and location of the temperature minima, first detected from space with Herschel, are well reproduced by the theoretical models of this paper. The temperature minimum for α Cen B is lower than for A and occurs at a lower height in the atmosphere, but for both stars, Tmin/Teff is consistently lower than what is derived from optical and UV data. In addition, and as a completely different matter, a third point source was detected in Band 8 (405 GHz, 740 μm) in 2015. With only one epoch and only one detection, we are left with little information regarding that object’s nature, but we conjecture that it might be a distant solar system object. Conclusions- The submm/mm emission of the α Cen stars is indeed very well reproduced by modified chromospheric models of the quiet Sun. This most likely means that the nonradiative heating mechanisms of the upper atmosphere that are at work in the Sun are also operating in other solar-type stars.