to the hospice, and that the college simply hosts it?
That being the case, you'll need to decide on a new host first. Once you've done that, upload a copy of the site to the new host's server and ask them to request the college to retag the name to their (the new hosts') name server. You'll need to ask someone at the hospice to send a fax to the college, confirming that they should accept the retag request when it arrives. Once the college retags the name there'll be a delay of about 24 hours (often less) while the Domain Name Servers (DNS) around the world update their databases. During this time the site will disappear from the internet.
Once the DNS databases are updated (an automatic process) the copy of the site that's on the new host's server will magically appear as the name resolves to their name server(s).
If the ownership of the site is in the college's name they'll need to initiate the transfer of registration to the hospice. This is simply a matter of completing a form - usually online - and waiting for the transfer process to complete. The procedure varies slightly, depending on the high level that the domain name is in (.com, .org etc.) but it's pretty straightforward, and full details of how to do it can be found on the website of the registering organisations, or in the documentation that arrived following the original registration.