Here are simple steps which which you can host your new domain with your current hosting account.
1)Create an account for your new domain with your unlimited hosting account. You can do this through webhosting control panel like cPanel or Plesk.
2)Once you are able to create an account, point your domain to your host's nameserver with the help of your domain registrar. If you are not sure how to do it they will help you.
3)Generally it takes 24 hours for dns records to propagate and update worldwide. Meanwhile you can upload data files into your new account.
4)Your domain will be live and ready for use as soon as dns records propagates.