Sony Corp. said computer hackers breached security for a second online service, gaining access to personal information for 24.6 million customer accounts as part of a broader attack on the company that has compromised data for more than 100 million accounts.

Sony Online Entertainment, a San Diego-based subsidiary that makes multiplayer games for personal computers, said it shut down its services Monday amid concerns a hacker may have accessed names, birth dates and addresses for its users.

The...