He had an operation on his hand in Edinburgh on Monday and surgeons said the ligament in his thumb had broken away from the bone.
Armstrong, 26, was expected to be out of action for eight weeks and will thus miss the crucial game against the French that will decide who ends up with the championship wooden spoon.
Scotland gained its first tournament point from the drawn game against the Irish in Dublin on Saturday in a match that failed to produce a try.
Two Eric Elwood penalties, the second nine minutes from the end, canceled out two successful kicks from Gavin Hastings who passed 450 points in international rugby in the process.
The French meanwhile remained anchored on two points after going down 18-14 against England at Parc des Princes on Saturday with Rob Andrew's boot supplying all the visitors' points.
Andrew kicked five penalties and a drop-goal as England, resolute in defense in the face of considerable French pressure, registered its seventh successive win over their opponents.
England led 9-0 at half-time and Andrew's drop-goal gave it a 12-0 cushion soon after the interval.
France fought back to within a point thanks to a try by Abdelatif Benazzi, later shown to have put a foot in touch as he went for the corner, and two penalties from Thierry Lacroix, but lacked the cohesion in attack to pull themselves out of trouble.
England now faces unbeaten Wales in a Twickenham showdown on March 19 and need to win by 16 clear points to lift the Five Nations trophy.
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