How can I nag only non-subscribers to subscribe to a newsletter?

My situation is this: I've got an existing website and email newsletter. I've been told that the newsletter signup on the site isn't prominent enough and to implement a modal popup that says something like "If you're enjoying this site, please subscribe to the newsletter!"

If you're already a subscriber, you don't need to see it and it would be annoying, especially every time you navigate to a page. But I don't know that about you, because I only know the email addresses of the subscribers, and not that of any anonymous web visitor.

I'm thinking I could show this popup but allow someone to say "I've already subscribed" or "Not now, thanks", and set a cookie so that, on that machine and browser, at least, they wouldn't be nagged again. But I'm not sure if this is a good work flow for existing subscribers, and maybe it's too easy to dismiss for non-subscribers.

What's a good UX design pattern for this situation?