  $\alpha$ 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.
