I'm not sure what a cornish hen is

I've never seen them sold here.
ETA - its what we call a poussin! (i.e. a young hen only a few weeks old, typically less than 2lb). You can treat it as you would any chicken - roasting time according to weight so for 2lb would be about 1 hour - but wathc out it doesn't get dry (baste or marinade well). There are lots of recipes onlione