The Brooklyn Heights library is neither the oldest nor the most dilapidated branch of the Brooklyn Public Library system. But the 52-year-old limestone building is nonetheless ripe for demolition.

It sits on land that developers crave, in a fashionable neighborhood where housing is in high demand. And so the library system, desperate for money to pay for $230 million in long-deferred repairs for its 60 branches, has embraced a novel financing model that is increasingly being used around New York City as a way to pay for government services.

The library, on Cadman Plaza, along with another library near the Barclays Center, would be sold to developers, torn down and then rebuilt at no public expense on the ground floor of a new apartment tower....