Alpha Centauri A is the closest solar-type star to the Sun and offers the best opportunity to find and ultimately to characterize an Earth-sized planet located in its habitable zone. Here, we describe initial results from an Atacama Large Millimeter/submillimeter Array (ALMA) program to search for planets in the {alpha} Cen AB system using differential astrometry at millimeter wavelengths. Our initial results include new absolute astrometric measurements of the proper motion, orbital motion and parallax of the {alpha} Cen system. These lead to an improved knowledge of the physical properties of both {alpha} Cen A and B. Our estimates of ALMA's relative astrometric precision suggest that we will ultimately be sensitive to planets of a few tens of Earth mass in orbits from 1 to 3au, where stable orbits are thought to exist.