It seems to me this would be entirely possible: just use any existing
browser and give it an extremely limited domain name server -- and
disable nslookup.... By adding a site to the DNS, you approve all its
pages (without having to approve each page individually) but not
necessarily all its links.
Any one student with net access outside of school could get around this
pretty easily, but as others have mentioned, there will always be a way
for a determined (or even not-so-determined) student to get around these
systems. At least this solution requires no new browsers, no extensive
work on filters, etc. In addition, any student who gets around it will
have to learn something about IP addresses and how to find them, so the
breach won't be a total loss. :)
Marc