BRUSSELS—Euro-zone leaders agreed Thursday on a new €109 billion ($157 billion) bailout for Greece and new steps to prevent its debt crisis from metastasizing across the Continent—in a plan expected to trigger the first debt default by a nation using the common currency.

The meeting also produced a stark and open-ended declaration: The wider euro zone is committed to financing countries that take bailouts—thus far, Greece, Ireland and Portugal—for as along as it takes them to regain access to private lenders.

The...