What you are asking for is certainly possible, and the most cost-effective solution that will work with almost any paid hosting solution you may currently have (no matter how cheap) is often called "Domain Cloaking". This is where pages are served from a resizable frame the size of the browser window, and is a feature offered for free by most domain registrars.
All other solutions which would be a lot more effective and would not lock your secondary site in a frame would all have additional requirements (e.g. a dedicated ip address and the ability to configure additional host-headers) which would probably increase the cost of your original site to the point that you might as well simply purchase a second hosted website with a budget-friendly provider (e.g. GoDaddy) to serve your second site for less than the cost of a single trip to your favorite caffeinated beverage establishment.