The 1988 champion, 59, starting at five over, dropped four shots on his front nine before a quintuple bogey on 11 and a double bogey on 14 left him 16 over.
Fellow Scot Russell Knox also missed the cut - a birdie at the last was no consolation as he ended eight over.
After an opening 76, Knox dropped four shots on the front nine before carding three straight bogeys from the 12th.
The cut came at six over, meaning defending champion Danny Willett will also miss the weekend after he finished seven over.
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Former winner Sandy Lyle missed the cut at the Masters after slumping to an 11-over-par 83, leaving him next to last.