Gmail: how to remove "on behalf of" from sender name

I use my Gmail account to send email from my philip@psteiner.com email account, but I noticed the recipient sees it as sent from me "on behalf of" my Gmail account, i.e.

From: dontusethisemail@gmail.com on behalf of Philip Steiner <philip@psteiner.com>

The tricky thing for me is that my psteiner.com domain is hosted on Google Apps, so I have no mail server available at the psteiner.com domain, nor do I want to pay bucks every month just for this service - at least not when Google can provide it for free.

Luckily Steve Mould over at LifeHacker was generous enough to share his solution for configuring Gmail to use Google Apps as the outgoing mail server. Huzzah!