Precision Millimeter Astrometry of the $\alpha$ Centauri AB System

$\alpha$ 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 (HZ). Here we describe initial results from an 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 10s of Earth mass in orbits from 1-3 AU, where stable orbits are thought to exist.