When she was a university student, Yuriko gave herself over to carnal desire.
Literal
University-student [genitive-の] time, Yuriko [subject-が] carnal-desire [to-に] body [object-を] entrusted.
A literary, novelistic sentence. 愛欲 (あいよく, 'carnal desire / lust') is a literary Sino-Japanese compound: 愛 (love) + 欲 (desire). 身を委ねる (み を ゆだねる, 'to entrust one's body to / to give oneself over to') is a set literary expression for surrendering to something — here, desire. The temporal phrase 大学生の時 ('when [she was] a university student') sets the scene. 有理子 (Yuriko) is a personal name.