I've had a word with him, but his attitude is that if he's leaving, why should he care? How can I persuade him otherwise? Normally, his behaviour would be a sackable offence, but I don't want to sack him and possibly cost him his next job. Can you advise?
A: From everything you tell me, he deserves to be sacked. It's not as if he's just demob happy; he knows exactly what he's doing (or rather, isn't doing) and glories in it. Your conscience should also twitch a bit on behalf of his next blissfully ignorant employer. If this employee can behave with such boorish irresponsibility under these circumstances, he's not going to be a long-term satisfactory hire for anyone.
Did you, I wonder, give him a reference? If so, you'd be entirely entitled to rescind it - and to send a copy of your revised version to his future boss. You may even have a moral duty to do something like this.
Anyway, tell your less-than-admirable and soon-to-depart employee what you have in mind - show him your re-written reference, if you like - and give him one last chance before sending it off. It really should work.
For safety's sake, check with your HR people and company lawyer; but I doubt if you're laying yourself open in any way.