Lets say you're working on a startup with someone for a couple of months and they start to lose interest and aren't pulling their weight. And it has come to a point where just talking it out isn't going to help.
What is the best way to sever ties and continue with the business solo?
Each of us has 50% equity in the business (by word, no actual paperwork at the moment). The app hasn't launched yet but the product is almost there to completion.
I still feel passionate about the idea and want to continue with it solo. But I can't just take the work and continue since it won't be legal (even though we haven't done any paperwork).
For starters, next time, you've got to get this kind of stuff in writing at the beginning, including how one person could buy out the business from the other person.
The second point I want to make is that it's possible that you're underestimating the other person's contributions. (Not that this necessarily applies to you, but I'm thinking about everybody else who comes along and reads this question as well.) The reality is, it's human nature to put too much emphasis on our own contributions and de-emphasize everyone else's. We're just too close to it, and we see all of the nitty gritty details of what we're doing, and gloss over everyone else's work. So you want to be sure that your analysis of their contributions is as objective as it can be.
But assuming you get past that part, I'd simply go to the cofounder and have a frank discussion. Tell him/her that you don't feel like they're pulling their weight. You might be able to sort out your concerns here, but it sounds like that doesn't apply here.
So you just say, "Look, this isn't working. I want to buy you out. I value that company at $X, which means your 50% is worth $(X/2). I'll buy you out for that much money, or if you think it's worth more than that, you can buy out my half for that much."
This is often called a shotgun clause (if it were written down in an agreement) and it's the business equivalent of, "I'll cut the cake and you pick the part you want." It forces you to be honest about what the business is actually worth, and gives him/her a reward for contributions so far.
This may not necessarily result in you having the company. Your cofounder may buy it from you instead. So you need to be prepared for that.
Putting you and your cofounder in a shotgun situation like this could have some drawbacks (read the "Potential Problems" section on that Wikipedia link). Depending on your specific situation, something else might work better for you.
But the general idea is going to be the same. You'll have to talk to him and say it's not working, and offer either him/her or you the door.
Loosing momentum is common when building a product (key is to meet in person often) and it can be turned around, it gets harder 1+ years after launch with poor financial performance. Perhaps your partner is just dealing with some life events. Loss of enthusiasm isn't the worst thing, your test as a co-founder is ability to bring your enthusiasm into the relationship and re-energize your partner. Are their skills still valuable to you and can you really launch and grow the product without them? Can they complete and launch the product without you?
It would help to know whose skills are more valuable and what the contributions are so far and what they can be in the future. If you need to replace your co-founder's skills than you are better off at least trying to salvage the relationship. You can try to draw up commitment agreement for post-launch and list all the things the 2 of you would do, just this exercise would show level of commitment and contributions you can expect from each other.
If you don't want to make it work, then before shotgun approach rbwhitaker suggests, try to have a conversation where you simply point out what work you were hoping your co-founder would do/complete before launch (or whatever upcoming milestone is) and ask if they think they still could. Be nice, but with the goal of creating pressure. If your co-founder doesn't meet those expectations they will feel guilty and hopefully realize they failed you. Negotiations are likely to go your way if they understand they can't take on the business by themselves.