Colo a box, and find a competent admin. 1000 concurrent users isn't small either by the way. You also completely neglected to mention what the site was written in, and what webserver you are using. There is a major difference if you have 1000 concurrent users on apache tomcat, or a simple php site.
Personally you are better off hiring someone in house to manage it, but do yourself a favor, and do your research. There are a lot of people in the industry who talk big and suck at what they do. If he cant look at what you have, and setup a proper infrastructure for it within 30 days, fire him. The other option, is look at the smaller companies to do managed hosting for you. They usually have a few racks in a DC they use.
Dont bother with these big companies like hostgator. Most of their employees are worthless. Not to mention the recent story about one of their admins backdooring their servers. The more reputable ones stretch their employees too thin, so unless you are a squeeky wheel you will get neglected.