Welbeck was hurt during the Gunners' UEFA Europa League draw with Sporting CP after falling awkwardly and coach Unai Emery confirmed that the England international had suffered a break.
The 27 year-old had surgery on Friday and has since been operated on for a second time.
The club is uncertain when Welbeck will be fit enough to return to action.
"Further to sustaining an injury last Thursday against Sporting CP, Danny has undergone a second planned operation on his right ankle," Arsenal said in a statement. "Both procedures went well, without complications. Danny will now recuperate and start his rehabilitation. It's too early to say how long this will take.
"Everyone at the club will now be providing Danny with the best possible treatment and support in order to make him fit again as soon as possible."