I have to transfer a website to a new IP address on the same server in order to install an SSL cert. Apparently it will take 4-24 hours for the DNS changes to propagate. I asked my host's support if I could serve the website to both IP addresses and avoid downtime, but they said that's not possible. I'm calling BS, it seems to me it should be very possible. So, is it possible? If so, how do I do it?

My server is a VPS running CentOS with Apache and cPanel/WHM.