Encouraged by the surprising turnaround, Ms. Zanella’s parents and two siblings, who also had been living in the United States illegally, presented papers late Monday to ICE, as the agency is known, turning themselves in and requesting some form of legal immigration status.

“It’s an opportunity we are going to take,” Ms. Zanella said in a telephone interview from Dallas. “It’s better than being in the shadows.”

The about-face by ICE in Ms. Zanella’s case is an example of the kind of action Democratic lawmakers and Latino and immigrant groups have been demanding from the Obama administration to slow deportations of illegal immigrants who have not been convicted of crimes. In particular, pressure is increasing on President Obama to offer protection from deportation to illegal immigrant college students who might have been eligible for legal status under a bill in Congress known as the Dream Act.

In an April 13 letter, the top two Democrats in the Senate, Harry Reid of Nevada and Richard Durbin of Illinois, asked the president to suspend deportations for those students. But short of that, the senators asked Mr. Obama to set guidelines by which those students could come forward individually to ask to be spared deportation and to obtain some authorization to remain in the United States. The letter was signed by 20 other Senate Democrats. The Dream Act passed the House but failed in the Senate in December.

Homeland Security officials have said their focus is increasingly on removing immigrants who are convicted criminals. That, in fact, is what an ICE official told Ms. Zanella in explaining the new decision in her case.