What is the best way to upgrade subversion to 1.6.13 on ubuntu 10.04 LTS?
Ubuntu 10.04 ships subversion 1.6.6 and important bug fix has been done for 1.6.13.
If you absolutely need 1.6.13 now, then I suspect you would either have to:
Both of these ways will probably be harder to maintain than necessary though.
What I would suggest would be to use the subversion ppa; its objective is to hold a (relatively) recent release of subversion. As of today, the version of subversion in that ppa does not fix the bug (CVE-2010-3315) that you are referring to (it is fixed by either 1.6.12dfsg-2 or 1.6.13). If you are interested about what is happening to the subversion package, you can take a look at that mailing list thread.
There's a ppa you can use for more up-to-date versions: PPA
But it's still v1.6.12