Where does the name of Pont-Neuf come from?

There's a bridge over the Seine in central Paris called Pont-Neuf, and I've always wondered about its name. "Pont" is French for bridge, and "neuf," strangely, can mean either the number nine or "new." Is this the new bridge, or the ninth bridge? Or something else entirely? For all I know there was an important French guy named Neuf and they named it after him.

Anyhow, I'm in the middle of reading (very slowly) Métronome by Lorànt Deutsch; it's a fascinating history of Paris that goes from Métro stop to Métro stop and spans a few thousand years. Interestingly, he addresses the name of Pont-Neuf! Let me translate the last paragraph there for ya...

"Our four current bridges are reconstructions from the 19th century, and be aware that Pont-Neuf, finished in 1607, is not the ninth bridge in Paris but the fifth, and it's not the newest but the oldest, you follow?"

Uh, maybe I don't follow; I still don't know why it's called Pont-Neuf! However, I just thought to try Wikipedia, and it's confirmed: the neuf means new, as it was the newest bridge at the time it was built. Whoa, wait a second, isn't every bridge the newest one at the time it was built? Oh man, now I'm even more confused...