WARSAW—A recording of a private conversation among Poland's top leaders over the nation's finances, leaked over the weekend, plunged the country into one of the worst crises since its center-right government came to power in 2007.

In the recording made by undisclosed people last year, Poland's central bank governor can be heard negotiating with a government minister over the terms under which the National Bank of Poland could help the cabinet as it struggled to keep the cash-strapped government's public debt within legal limits.

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