Here are some details missed by other answers. This answer covers doing it right, and not stealing someone elses IP.
First, the vast majority of IP are already purchased. Second the BGP router protocol doesn't allow for blocks less than /24 or 254 usable IP addresses to be entered by universal agreement. Probably ICANN or similar standards committee.
First you need to see if your ISP is even willing to do this, and how much they are going to charge because its definitely going to be extra. EXTRA per month.
So, you have a couple choices you can buy a unowned /24, or you can try and purchase the /24 from someone who owns a larger block. Given how precious IPv4 addresses are today you might have to pay a small fortune to get a specific IP and its going be worst because you need a /24. Also note big company's like HP,google,microsoft won't be selling any of there's.
Pretend you wanted and got 1.2.3.0/24. Your ISP has to host it, and have it entered into the BGP routing table. Then it will work. The fees for all of this are going to be high and monthly. I got 8 ip for $15/month, many are happy to charge $15/month for 1 IP.
I can not stress enough the high cost of this, and its going to take a bunch of time and effort to set it up.
Think about it this way, say you wanted phone number 111-111-1111. If a random citizen owns this number, and you offer them a crazy amount of money like $100,000 they will probably release it you. Once you own it you can pay a service like MagicJack or Google voice or etc to host it for you.
This situation changes dramatically with IPv6.
There are 2^128 IP vs 2^32 IP (only 4 billion). So a fraction of the IP space is used right now and its much larger. So finding a unused IP you like will be much easier. The BGP routing tables for ipv6 require a /48 before they create an entry for it so you will have to buy that many.
However, not all ISP are handing out IPv6 to customers so access to ipv6 is limited right now. There are passthrough services, but now you have to set that up and pay them an extra fee in addition to all the other fees you are going to have to pay.