It all depends on your local labor laws. Does the employee have a contract (written or implied)? If so, is the employer/employee relationship specified as at-will? What does your employee handbook say regarding termination?
if you hadn't picked up the poor performance before it got to this stage then you really do need to review your own management style.
I'd have to agree with this- if you haven't let the employee know before now that the mistakes are an issue, you're definitely asking for trouble.