The former Dundee United wide man has joined on 93-day short-term loan - running up to, and including, the match at Middlesbrough on 14th December.
Albion head coach Oscar Garcia commented, "We were keen to bring extra options, and with Andrea now ruled out for a longer period than we initially expected, another wide player became a priority.
"Craig is very experienced at this level and knows what it takes to be successful in the Championship. He can play on either wing, or up front as a second striker; he also has international experience and is a player who will improve our squad."
After progressing through the youth ranks of his hometown club Ayr, Conway joined Dundee United in 2006, going on to serve the club for five years.
While with the Tangerines the Scot made over 100 appearances, including in the 2010 Scottish Cup Final victory over Ross County where he scored a brace.
Conway left his native Scotland the next summer to join Premier League hopefuls Cardiff City in the Championship and made over 50 appearances and scored seven goals for the Welsh outfit; including the strike that clinched the 2012/13 Championship title as the Dragons fulfilled their top-flight ambitions.
The 28-year-old was called up to his national side for his debut in 2009 against Japan, and most recently came on as a second half substitute against England in the 3-2 defeat at Wembley last month.