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As you have been told by the other posters, it is very dificult to identify the IP address of many users.
Often a users ISP can mask the address so that you just get a public IP the blocking of which might stop many users of getting to yur site, or the ISP will probably (as is the case for most ISPs) assign the users IP address dynaically, therefore stopping the address this time will do nothing when he gets a new IP assigned from his ISP.
So how to get past the problem?
First of all you can make it so that only members can post on your site, this way you will get the member name of the article poster, then you can boot them from your site, they might sign up again, just boot them again, they will get bored and move on eventually.
Remember to make new members activate their accounts, this way the user is going to have to get another email address everytime he wants to sign up and post junk, again this will help to make him bored so that he goes away.
And I guess another option might be to edit the templates of the modules that this user is abusing so that the IP address is captured from each user that submits to the site, you can ban the IPs this way, but as stated before, most IPs are dynamic, which means you might end up with a really big list of IPs to block. (possibly even set it so that the IP address is only displayed to admin level users)
I suppose once you have the IP, then if you wanted to take the risk you could perhaps use a .htaccess file to ban all address's from that ISP, but that might well cause you more problems than it will cause him.
Personally I would go for restricting access to modules to members and making new members activate their accounts, that way they have to provide a real email address, or at least one they can check (Its not fool proof, there are sites that will help you get past email activation, but it might help make the process of posting empty posts a little less fun).
Dave