Yes, you can do that. It is called colocation. Essentially you provide the server and the colocation provider supplies everything else: power, cooling, security, and in some cases they provide bandwidth in some cases you can provide it yourself.
They will base the cost on how much physical space your server takes up, how much power it uses how much heat it generates and if you need bandwidth, how much bandwidth you will be using. Typically if they provide bandwidth they will charge you a fixed amount for a certain number of IP addresses and a set CIR (committed information rate). You can either pay for a fixed amount of bandwidth or you can pay based on usage. If you pay for fixed bandwidth then they will give you that amount and you will never be able to use more. It is quite common for you to buy burstable bandwidth though. In this scenario they will provide you with a port that can go up to 100 Mbps for example and bill you based on your average utilization. Typically this is done using the 95th percentile billing model (Google can explain it better than I can). So they may charge you $50/mo per megabit of 95th percentile bandwidth so if you average 10 Mbps then you would pay $500 for that month.
Having said that, if you only have 1 server then it would probably be drastically simpler and easier to either rent a dedicated server or use a virtual server (VPS). Companies such as Rackspace and Amazon Web Services provide virtual servers. In that model you would pay for the virtual server based on how much CPU, RAM and disk you need. You also pay for bandwidth but in this case you pay based upon how much data you transfer, not your average utilization. For example, AWS charges about 10-12 cents per gigabyte of data your server sends to the Internet.
There are other advantages to using a virtual server. You no longer have to worry about hardware failures since your server is a virtual machine if the host it is running on has a hardware problem it can easily be moved to another host. Additionally it is easy to upgrade the virtual machine to have more or less CPU/RAM/Disk based upon your usage.