Their conversation went on and on until late into the night.
Literal
Night [subj-が] deepen-[until-まで] their conversation [topic-は] did-not-run-out.
話が尽きる ('the conversation runs out, topics are exhausted') is a set phrase. The negative 尽きなかった ('did not run out') describes conversation that kept flowing. 夜が更けるまで ('until night grew late') marks the duration endpoint.