There are a number of approaches. One is to set up your favorite redirector (mine would be *bsd/pf, second *bsd/ipfilter) listening to the old address and forwarding to the new address.
Yes, there are a number of way to skin this cat, and they can make a net admin who comes upon an in-process migration where folks hadn't remembered what Dave said about the DNS feel like a hero when it gets pulled off. However, the sad result is that said net admin gets the wrong reinforcement and now is also lliable to ignore what Dave said about the DNS in future migrations and is setting themselves up for great trouble. For there be dragons lurking in many of these port redirection/tunneling/aliasing tricks.
Now, there exist some resolvers out there, not so many as before, but still a few, which do not honor the TTL. And there used to be some browsers which cached their names instead of relying on their resolvers. In both of those cases, Dave's sage advice would *seem* to be incorrect. However, it would quickly identify said resolvers and browsers, and allow for their expeditious removal and replacement. Thus proving for all time the ultimate value of what Dave said abour the DNS.