You CANNOT simply sack the guy if you want to stay within the law. However if you wish to step outside of the law the the most your employee can claim successfully from you is whatever period of notice he is entitled to - either by contract or by statute.
He may not be able to bring a claim of 'unfair dismissal', but if you ignore your obligations/his rights, he would have grounds to raise an action against you for 'wrongful dismissal' = breach of contract.
Under S.1 (4)(e) of the Employment Relations Act 1996, you are obliged to provide all of your employees with a written statement of the required period of notice - this period applies to both the employee and the employer. In addition, S.86 of this Act, provides the statutory minimum periods of notice required. For your employee this is one week - since he has been employed for more than one month, but less than two years. However, if you have provided a written statement of the amount of notice required (as you are legally obliged to do), and it is for one month, then you MUST observe that. You cannot give him just the one week's notice.
You can only sack someone on-the-spot (summary dismissal), if the employee commits an act of gross misconduct - such as dishonesty or violence or sexual harassment. In that case they'd be entitled to nothing.
So the good news (for you) is that you CAN fire this employee - for any reason - so long as you serve notice, and observe the appropriate period of notice.
I hope in light of this advice though, that you do indeed treat him well - better than he may be entitled to even, as you have indicated. If nothing else, your other employees will be more likely to show trust and loyalty to you (and work harder) if you show both toughness (by dismissing him), and some compassion (by giving him a bit more than he may in fact be entitled to). If you just fire him, how can you expect any loyalty from your