LONDON — Britain’s international development secretary, Priti Patel, on Wednesday became the second cabinet minister to quit in a week, a significant new setback for Prime Minister Theresa May as her shaky government grapples with critical decisions ahead of the country’s departure from the European Union.

Ms. Patel, 45, resigned after breaching ministerial rules by holding a dozen unauthorized meetings with Israeli officials during a summer vacation, giving a misleading impression about whom she had informed and failing to disclose all the relevant details to Mrs. May.

This new political turmoil is likely to increase speculation about the strength of Mrs. May’s grip on power, and comes at a tense moment in discussions on quitting the European Union, or Brexit. The resignation also follows a series of sleaze and sexual harassment allegations in Parliament, which prompted another cabinet minister, Defense Secretary Michael Fallon, to quit last week, and threatens another.

The litany of accusations against Ms. Patel — who was summoned home from a working visit to Africa — made the minister’s resignation almost inevitable, turning her slow-motion departure from government into a protracted political pantomime.