The young Bridgend hooker sustained the injury in the opening minutes of Wales' 30-22 defeat at Murrayfield.
"I fractured the bone within the first 10 minutes of the game as I tackled Bryan Redpath. It hurt like hell but it wasn't affecting my game so I just got on with it.
And he remains hopeful he will be ready for the game against France on March 29.
"Ireland is an outside hope at best but I am hopeful that I can be fit again in time for France."
The hooker was handed his first international start after skipper Jonathan Humphreys was ruled out with a shoulder injury.
"Starting for Wales was something I had always dreamed about," he said.
Williams was fast-tracked into the Wales A set-up last season and made his Test debut as a replacement for Mefin Davies in the defeat to Italy last month.