Context. The precise mechanisms that provide the non-radiative energy for heating the chromosphere and the corona of the Sun and those of other stars constitute an active field of research. By studying stellar chromospheres one aims at identifying the relevant physical processes. Defining the permittable extent of the parameter space can also serve as a template for the Sun-as-a-star. This feedback will probably also help identify stars that potentially host planetary systems that are reminiscent of our own. Aims. Earlier observations with Herschel and APEX have revealed the temperature minimum of α Cen, but these were unable to spatially resolve the binary into individual components. With the data reported in this Letter, we aim at remedying this shortcoming. Furthermore, these earlier data were limited to the wavelength region between 100 and 870 µm. In the present context, we intend to extend the spectral mapping (SED) to longer wavelengths, where the contrast between stellar photospheric and chromospheric emission becomes increasingly evident. Methods. The Atacama Large Millimeter/submillimeter Array (ALMA) is particularly suited to point sources, such as unresolved stars. ALMA provides the means to achieve our objectives with both its high sensitivity of the collecting area for the detection of weak signals and the high spatial resolving power of its adaptable interferometer for imaging close multiple stars. Results. This is the first detection of main-sequence stars at a wavelength of 3 mm. Furthermore, the individual components of the binary α Cen AB are clearly detected and spatially well resolved at all ALMA wavelengths. The high signal-to-noise ratios of these data permit accurate determination of their relative flux ratios, i.e., S B ν /S A ν = 0.54 ± 0.04 at 440 µm, =0.46 ± 0.01 at 870 µm, and =0.47 ± 0.006 at 3.1 mm, respectively. Conclusions. The previously obtained flux ratio of 0.44 ± 0.18, which was based on measurements in the optical and at 70 µm, is consistent with the present ALMA results, albeit with a large error bar. The observed 3.1 mm emission greatly exceeds what is predicted from the stellar photospheres, and undoubtedly arises predominantly as free-free emission in the ionized chromospheric plasmas of both stars. Given the distinct difference in their cyclic activity, the similarity of their submm SEDs appears surprising.