僕がいない間、この赤ちゃんをお願いするよ。

、。

Please look after this baby while I'm gone.

Literal

I [subject-が] not-exist during, this baby [object-を] please-entreat [emphasis-よ].

僕がいない間 ('while I'm not here') uses 間 ('interval, duration, while') as a temporal noun. この赤ちゃんをお願いする is a casual colloquial expression — 'please do this (baby)' where お願いする can take an object for entrustment. Very casual babysitting request.