One common way for popular sites to do this is to invite visitors (to the existing site) to try out the 'new and better' site. That way you put users in control - and if they don't feel comfortable with the new, they have the option to go back to the original - but you have to make it clear they can do this!
Twitter for example did this - 'Try new twitter' - when they moved to a proper two column content.
If you have the opportunity to do this, it's a useful risk-managed way to gradually engage visitors in a controllable manner, and you can be flexible about the overlap period in case of problems on the new site or unexpectedly slow take-up. It's also relatively easy to A/B test different ways of inviting users to try out the new site.