The rain arrived just as the players had started their warm-ups and it continued until early afternoon, by which time pools of water were left standing on the outfield.
With further heavy rain forecast, umpires Neil Mallender and Tim Robinson decided to abandon play for the day after a 1.40pm inspection.
The washout continued a dismal summer of international cricket at Edgbaston where three days of England Test against the West Indies were washed out completely along with a one day-international against Australia in July.
Australia A will resume their first innings on 308 for nine when the match continues on Thursday after Glamorgan seamer James Harris took six for 102, his best figures since May 2007, on the opening day.
Australia A are playing their last game of a four-match tour and are seeking their first win after a defeat by Durham and draws with Derbyshire and the Lions in last week's first match at Old Trafford.