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 most literal translation of your sentence would be:
Ich gebe ihr/ihm den Hörer.
And people do say that even with wireless phones that have a base station somewhere. For a cell phone. There, you would just pass on the "Handy". The suggestions of the other answers are fine as well.