DUBLIN—Ireland's troubled banking system became the latest flash point in Europe's continuing economic crisis, as the government said it would split up the weakest of its major banks to stave off a run by depositors.

Irish Finance Minister Brian Lenihan, days after meeting European Union officials, said state-owned Anglo Irish Bank Corp. would be divided into a government-backed bank that would hold customer deposits and an "asset recovery bank" holding the bank's increasingly bad loans. The asset-recovery bank could be sold...