Usually the three-warning system (verbal, written, final written) is done to provide documentation to prevent unemployment collection or have evidence against a claim. While verbal warnings are just that - verbal - you would still document them on paper.
With a warning system, you would need to allow enough time (usually 1-2 months) between each warning to allow time for improvement to CYA. If they are still unable to to meet the expectations set/discussed, you'll have evidence to refute a possible unemployment claim.
If you've never had anyone else collect unemployment and you've been paying into unemployment insurance, I wouldn't worry about just firing them when you want to - at least not for this first claim. When we've let people go, whether they've been on corrective action or not, we give two weeks pay and send them on their way (don't want someone in-house who knows the end is near). 3-6 months is a long time to wait for a small business, and they can still file (and you would have to contest their claim - sometimes a lengthy process. Since the recession started, companies contesting unemployment claims has skyrocketed). Unemployement insurance rates (at least ours - US-based company) are based off a 5-year rolling period of claims vs. total payroll.
I'd check with an employment attorney before making any final decisions. For us, the first claim was fine - the second one, however, was the painful one. If your payroll has changed substantially in the timeframe since your unemployment insurance was first issued/quoted, you might be in for an unpleasant surprise. My completely non-legal advice would be to cut this person loose and employ a more rigid corrective action system in the future.