The word you're looking for is "der (Telefon)hörer". The literal translation is "the listener", but in context of telephones it was the word for the "bone".

> Ich lege den Hörer auf.  
> I put down the receiver.

> Ich nehme den Hörer ab.  
> I pick up the receiver/the phone.

The sentence you're looking for would be

> Ich gebe ihr/ihm den Hörer.

You could still say that for wireless phones that have a base station somewhere, but it sounds a bit out of place for a cell phone. There, you would just pass on the "Handy".