。
She let out a sigh of relief.
Literal
She [topic-は] relief [of-の] sigh [object-を] let-leak.
ため息をもらす ('let out a sigh') uses もらす ('let leak/escape') instead of the more common つく ('attach') in ため息をつく. もらす frames the sigh as something that slipped out involuntarily, with a slightly more literary/considered feel. 安堵 (あんど) is a Sino-Japanese noun for 'relief, easing of anxiety' — more elevated than 安心 ('feeling at ease'); the 安堵のため息 collocation is fixed in narration of moments of release.