It could also upset the balance within Mrs. May’s top team, which has been constructed to include enthusiasts for Britain’s departure from the European Union, or Brexit, and those, like Ms. Rudd, who opposed withdrawal during a referendum in 2016.
For days Ms. Rudd had resisted pressure to quit, but the onslaught intensified last week after she told a parliamentary committee that there were no targets for deporting illegal immigrants.
That statement proved to be incorrect and, though Ms. Rudd later insisted that she did not know about the targets, this defense crumbled, too. On Friday she admitted in a Twitter post that she had been copied on an email that mentioned migrant deportation targets, but she insisted that she had not read it.
Another document leaked on Sunday appeared to show that she had discussed deportation targets in a letter sent to Mrs. May.
Having misled lawmakers, Ms. Rudd found her position increasingly untenable and ultimately opted to quit rather than suffer another ordeal on Monday, when she was scheduled to appear again before Parliament.
In her resignation letter, Ms. Rudd acknowledged that she had “inadvertently misled” lawmakers over deportation targets.
“I should have been aware of this and I take full responsibility for the fact that I was not,” she wrote.
Mrs. May said that Ms. Rudd had answered questions from the parliamentary committee “in good faith” and that she was “very sorry” to see her leave.
Some had predicted that Ms. Rudd would survive the scandal because she was acting as a political shield for Mrs. May, the previous home secretary. During her lengthy tenure in the job, Mrs. May tried, and failed, to bring net migration into the country below 100,000 people a year.
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Though members of the Windrush generation had a right to be in the country, many struggled to prove it, becoming ensnared in a system that sought to make life hard for those in Britain illegally.
The scandal has also raised concerns about whether European Union citizens, currently living in Britain, might face similar bureaucratic problems after Brexit.
Unless the crisis can be calmed quickly, Ms. Rudd’s departure leaves Mrs. May more exposed to criticism from the opposition Labour Party and others.
“After this scandal and its botched cover-up, Amber Rudd’s resignation was inevitable. It should have come sooner,” Diane Abbott, the shadow home secretary for Labour, said in a prepared statement.
“The architect of this crisis, Theresa May, must now step forward to give an immediate, full and honest account of how this inexcusable situation happened on her watch,” Ms. Abbott added.
Ed Davey, who speaks for the centrist Liberal Democrats on home affairs, said that it was “clear that Amber Rudd has ended up, at least partly, being the fall guy to protect the prime minister.”
“Theresa May must face questions now given these dreadful failures largely took place under her watch as home secretary,” Mr. Davey added in a statement.
Ms. Rudd’s departure also removes from the cabinet one of its strongest pro-European voices. But she will remain a lawmaker and might strengthen the number of Conservative Party backbench lawmakers prepared to defy the government if it pursues a “hard” Brexit, or clean break with the bloc.
Since Mrs. May lost her parliamentary majority in elections last June, her leadership has often been under question, and she has already suffered three prior cabinet resignations.
In December, Damian Green, who was effectively Mrs. May’s deputy, quit after an investigation found that he misled the public about pornography found on his parliamentary computer.
Priti Patel, the international development secretary, also resigned from the cabinet last year after breaching ministerial rules by holding a dozen unauthorized meetings with Israeli officials during a summer vacation, and a scandal over sexual harassment in Parliament prompted the departure of Mrs. May’s defense secretary, Michael Fallon.