His first match in charge will be Wednesday's league game at Bologna which was abandoned because of snow last month.
Ranieri resigned following fan protests and after eighth-placed Roma went down 4-3 at Genoa having been 3-0 up in their Serie A match.
What happens after that is open to debate with Chelsea coach Carlo Ancelotti being linked with a return to Italy by media in both England and Italy unless Chelsea's slump in form ends.
Ranieri's departure has come at a difficult time for Roma, who did well to finish second under him last season, but have slumped in the standings after a poor run.
The club's supporters have been protesting violently, but the fans' feelings are unlikely to put off an American consortium from trying to buy the indebted club, Rome mayor Gianni Alemanno said on Monday.
"I think we are talking about a consortium made up of experts who won't be scared," Alemanno told reporters.
Italy coach Cesare Prandelli has praised Ranieri for showing "responsibility and dignity" by resigning - a rare step these days by a coach as members of his profession usually wait until they are fired, collecting a bumper payoff when their time is up.
