Ah, we have been here - about 7 months ago. My son's first girlfriend - it lasted 5 months - she broke up with him once to see another boy - he took her back - she dumped him 3 months later.
He was 15.
He was heartbroken.
All of it played out very publicly in school and on FB.
He did not get a bad reputation from it. In fact, most mutual friends supported him. Um, kids he didn't even know supported him.
There is less embarrassment than you realize - this is high school. Teens are crazy and emotional and couples pair up and break up daily. This is, unfortunately, normal for them.
You let the genie out of the bottle when you let him get a girlfriend. Now, you need to support him and help him navigate his way through. You cannot do this by forbidding him from seeing the girl - no quicker way to make him dig in his heels and continue dating her long after he wants to.
You don't turn your head away either. You keep an open line of communication with him and you begin by telling him you were wrong to forbid him to contact her. Yup, suck it up.
Oh, you definitely tell him that 2 hours phone calls to anyone are forbidden when he is babysitting and punish him for that, but you do not forbid contact with her. It is pointless and will create resentment between you and your son. And your relationship with your son will last much longer than his relationship with this girl.
Instead you talk to him about dating - tell him anecdotal stories about your dating life. You know, "So I dated this guy once, and he.... boy, that made me feel.....so I broke up (married, whatever) him."
Lead by example - teach him how to be in a relationship and how to handled this new phase of his life.
Set ground rules and limits, like you do with other activities, but give him some headway on this.
It is soooo not easy - but our parents made it through it and so will we.