When President Obama signed a new federal education law to replace the much-reviled No Child Left Behind last month, there was a whole lot of cheering in public education circles.

But they weren’t the only ones celebrating. Advocates for private schools also were pleased.

As it turns out, deep within a law usually debated for its effects on the nation’s public schools are provisions that dictate what resources private schools will receive to serve their most vulnerable kids and train their teachers.

Those lobbying for private schools on Capitol Hill say the newest version of the Elementary and Secondary Education Act — dubbed the Every Student Succeeds Act, or ESSA — restores what they say was the intent of the original 1965 law: Poor kids should get the extra help they need, regardless of where they go to school.

“We’ve returned to the equity provisions that were part of the Elementary and Secondary Education Act since 1965,” said Sister Dale McDonald, director of public policy and educational research for the National Catholic Educational Association.

The law says that school districts have to equitably share certain services with private schools, including, most prominently, Title I services for children from poor families. So if a school district spends $700 providing Title I services to each of its low-income children, it is supposed to pay the same amount for services to children in private schools.

But No Child Left Behind chipped away at that promise, advocates say. Instead of sharing the entire Title I allocation with private schools, school districts were allowed to set aside substantial portions of it for their own use. And so the shared pot shrank, and private schools received less.

It’s not clear how much less they received nationwide: The National Catholic Educational Association said it does not keep such figures nationally for its own schools, and there also are no reliable data on service reductions at all private schools nationwide, according to the Council for American Private Education.

But there are signs that private schools in some communities saw a change.

In the Archdiocese of Washington, for example, the rule changes under No Child Left Behind meant that the number of children receiving Title I services — such as extra instruction in math or reading — fell by more than half since 2000, from 848 to 349. Baltimore’s archdiocese has seen a similar rate of decline, a rate that Catholic officials said can’t be explained by falling enrollment alone.

Setting aside Title I dollars for their own use was a legal way for school districts to boost their coffers at a time when many were strapped for cash. “It’s not like they were cheating anybody,” said Joe McTighe of the Council for American Private Education. “But we thought that it was not fair and was against the original principles” of the 1965 education law.

ESSA, the new version of that law, won’t allow the Title I set-asides. It also tightens rules for Title II, or teacher professional development, in a way that advocates believe will result in greater access to federally funded teacher training.

The new law also requires state education departments to identify an employee who will serve as an ombudsman for private schools, tasked with making sure that school districts are meeting the letter and spirit of the new law.

“ESSA has stronger provisions in it for the equitable distribution of services, and that’s important because the students in Catholic schools, they’re the public,” said Thomas Burnford, interim president of the National Catholic Educational Association.

“Catholic schools serve the public and that public pays taxes and those taxes pay for services through ESSA,” Burnford said. “There’s a right on the part of private schools to have real access to those services.”

The NCEA has a rundown here of the legislative language that describes how resources should be shared with private schools.

ESSA mandates that school districts share federally funded services with private schools, but money does not necessarily have to change hands. A school district might provide one of its own employees to teach remedial English instruction at the private school, for example.

Many private school advocates would have liked to have seen ESSA embrace Title I “portability,” a policy akin to vouchers that would allow federal dollars to follow a child who enrolls in a private school. But that would have been politically impossible in a divided Congress whose leaders wanted very much to reach a compromise that Obama would sign.