you have two options. a redirect host, or a proxy. because of ssl, because of ssl updating the came will not work. also often host headers are used for routing.
for a redirect host, the Https://MyNew_Internalsite.MyOrg.com host needs to be configured to do redirects. how this is done is based on the host. typically a load balancer has this feature, but you can use IIS url rewrites, a static site with hen redirects, or an application.
a proxy. again a load balancer can be used. you will probably need to modify the response (typically with sed scripts) to fix cookie paths and imbedded urls.