I feel your pain — I've been there several times in a couple of my companies. Each situation ended up being unique, and had to be handled differently.
I think there are a few things to consider before you make your decision:
1. What is in your cofounder's way?
Is you cofounder being held up by a lack of clarity? Lack of motivation? Lack of autonomy?
One of my past cofounders was very good at getting the job done, but didn't naturally have the skill to lay out tasks in a manageable way. To get around this, I worked with the whole team (4 people) to write up process documentation that removed the need to "figure out what to do next" that was tripping up this cofounder.
2. What job was your cofounder brought on to complete?
And is it being completed?
One of my companies brought on a cofounder simply to give us a marketing platform — he had a huge online audience — but he did nothing else. At first, this caused tension; once we had specifically laid out who was on the team and for what purpose, it was easier to identify where responsibilities lay.
3. Is your cofounder capable of doing the job?
One of the more painful ordeals I've gone through in business is bringing on a good friend, then realizing that — despite his talent and intelligence — he just wasn't able to perform the job I'd hired him for.
His skills were better suited for a different job: he needs hands-on management; he works better with repetitive tasks that don't require big-picture thinking; he lacks assertiveness and confidence, which were critical for the management-level role he'd been hired to do.
After I tried to clear everything in his way, it became clear the company couldn't survive if he remained on the team. I had to lay him off.
4. Do you just simply not like the way this cofounder works?
In one of my startups, there was a cofounder who I didn't know all that well, but he had amazing industry contacts and domain knowledge.
However, once we started working together it became clear that we had VERY different working styles. He drove me completely nuts with (what seemed to me to be) a very ADHD-style of planning, with projects starting and being dropped and then coming out of nowhere with a call at 21:00 to discuss something critical that would be forgotten tomorrow.
I'm sure I drove him nuts, too.
So eventually we ended up selling that company — it was that or shutter it — because we knew there wasn't a chance we'd be successful if we continued as we were.
Working with other people is tricky in general. Our instinct is to assume that we're the best workers on the planet and everyone else is incompetent, an idiot, a slacker, or all of the above.
Usually it's a combination of an organizational-level lack of clarity, poor communication, no processes, and (sometimes) plain ol' we-don't-see-eye-to-eye-on-things-ness.
Hopefully that helps. Feel free to get in touch if you'd like to hear specifics on my situations, or if you'd like any help devising a strategy for resolving your cofounder trouble.