eCart 6 has an improved way of doing it as well... What you will have to do is this:
1) on your cart page, remove the checkout redirect server behavior... a simple redirect won't carry the cart over to the new site
2) on that same page apply the server behavior: eCart->General->Pass Session to Site
3) on your checkout.php page apply the server behavior: eCart->General->Recieve Passed Session
That should allow you to pass your entire session including cart contents to the new site so that you can proceed with checkout.