、。
While reading a book, he fell asleep.
Literal
Book [object-を] read-[ongoing-ている]-[while-うちに], he [topic-は] slept-[regret-completion-てしまった].
うちに ('while, during') expresses an unspecified span during which a situation holds; here the action of falling asleep happened at some point within the period of reading. The topic 彼 ('he') is introduced in the second clause because the first clause sets up the background context. The casual contraction 読んでる (= 読んでいる) is standard in spoken Japanese.