Quick changeover to new ISP

[Ned Trilby]

Hopefully someone can suggest a straightforward solution to my problem
of recovering a website using a new ISP.
This is my scenario: my customer runs his website "www.abc.com" from
his own site with ISP "ACME ISP". If his building burns down he comes
to my site and restores his website as normal; but my ISP is "XYZ ISP"
and I believe it can take up to 48 hours for the the route to the
website through the new ISP to be widely known.
Is there any method where I can have a backup web name to ip address
resolution in readiness at the "XYZ ISP" which can kick in as soon as
the machine is plugged in ?
TIA, Ned