Company wants a resignation letter

Hmm. You have verbally tendered your resignation.  In my view, if you want to leave the company in as professional manner as possible then why create hard feelings?  Further, since your resignation was not immediate and you provided (verbally) an end date then the company has obligations to calculate accumulated vacation pay and/or other benefits related to your employment.
There are perfectly legitimate reasons for the company to ask for your final date in writing. Assuming you are leaving on good terms then honoring reasonable requests preserves a positive relationship.

 You can refuse to give a letter, but you do this at your own risk. I say this assuming you want to stay there until your requested end date. 
The real question may be why you do not wish to provide this. If you want to keep on good terms with this company, and you have not resigned under duress, what would be the harm in providing a letter of resignation?

Thanks for the response guys!
 
I do not want to provide a letter because there is not a legitimate reason for them to ask for one. They can calculate time and pay based on verbal notice.
 
Since I posted this comment they have also asked that I make the letter out as to provide 2 weeks notice.
 
I previously told them I would be willing to work until April 30 and they have requested that  write a resignation letter stating I will leave in two weeks. I have no intention of doing this and want to know how others have navigated similar situations in their careers.

I don't understand why you wouldn't provide a letter and I would consider it unprofessional not to do so.   I agree with smacquarrie - the real question is why don't you want to provide this.
It doesn't need to be fancy - just restate what you verbally told them and sign it. 
Dear X, This letter is to advise you of my resignation effective April 30, 2012.  Sincerely, Y
That's all you need, and if that's your position why wouldn't you provide it?
Any other issues about the company wanting a different date are separate issues and should be dealt with as they arise.   You owe them at least a written confirmation of what you have already verbally promised.  
I don't know the particulars of this situation but there doesn't sound like there's a lot of trust and respect going on here.  But try to go with as much professionalism and grace as you can.   Take the high road - it's a small world.

Write a letter simply saying:
"To confirm the conversation with "boss", I'm resigning my position with "Company XYZ".  I appreciate my time here and the opportunities Company XYZ gave me.   My last day will be April 30th or an earlier agreed-upon date." 
The company has several legitimate reasons for asking for one.  Here's an obvious one: They want to have proof that you resigned to ensure they're not on the hook for unemployment benefits should your new opportunity fall through.  
By saying no, you're making a mountain over a mole-hill.  It just seems petty and unprofessional. 
 

 It is more than legitimate to confirm dates in written form. You need do nothing more than express that you have resigned with an effective date, no reason needs to be given, that is your business.
What you just stated makes it even more ridiculous that you are pushing back, they've interpreted you leaving with a minimal notice period and you have no intention with complying with that either? And when they stop paying you after two weeks, do you plan on pursuing legal action on the basis that you verbally provided a different date and refused to comply with a reasonable request to put it in writing?
You should re-read what you've written here from a different point of view, a reasonable person will come to the conclusion that a simple letter is a no brainer.