I rented some server space from a new host and have a static IP address and a domain name that I previously purchased, but I do not know how to register the domain name to point to my server.
My previous host did this for me, but my new host only offers hardware, i.e. they won't do it for me.
How can I register a domain name I own to point to my server's IP address?
You'll need to use a 3rd party DNS service such as DtDNS.com (which I own), ZoneEdit.com, DynDNS.com, or DNS service provided by your domain registrar. Essentially what you'll need to do is establish the service with a DNS provider, set the records to use your static IP address, then change the DNS server delegations at the domain registrar to use the provider you select. This will point your domain to your IP address.
Get a DNS provider.
Also, your title asks something different than your question. Both are separate operations, with the former needing a PTR record and the latter using an A record.
I would be wary that the host is unwilling or unable to give you help. I would be tempted to find another host, because this indicates to me that they won't be helpful when you have real problems later.
There are a couple of steps to accomplish the task ..
1- Get DNS setup for your domain. As you found out, some hosts do it, other don't. Some domain registrars do this, or you can use a third party such as Justin mentioned. This is where you actually map "samozdomain.com" to the server IP. As Ignacio hinted, the server itself is a subdomain, so you may want an A record for "servername.samozdomain.com" at the same IP, but this isn't necessary for web serving. If you are running email on this domain, you need to also have an MX record for the domain pointing to the email host.
--> The DNS provider should be able to help, particularly if you choose one already on this site!
2- Go to your registrar and point the "name servers" for your domain to the DNS service you just setup. The specifics on how to do this vary by registrar. The DNS service will have the correct settings somewhere clear.
At this point, the change is done. It will take some number of hours for the change to propagate through all of the caching on the internet DNS system, so users may end up at the old server for a day or so.
BTW .. here is a comprehensive question/answer on moving a web site between hosts.