Because I'm updating the backend (the database) I had to do a dump of my repository, perform the upgrade and then import the dumps into a new repository as described here. For example, to dump my Synergy repository I used a bunch of commands like this (where ~/Developer/svnrep/ is the folder where I keep all my repositories):

# stop anything that might be accessing the repository
sudo /usr/local/apache2/bin/apachectl stop


# create a temporary working space for the dumps
mkdir dumps && cd dumps


# perform the actual dump
sudo svnadmin dump ~/Developer/svnrep/Synergy > \
                   Synergy.dump


# move the repository out of the way
sudo mv ~/Developer/svnrep/Synergy \
        ~/Developer/svnrep/Synergy.old

Once I had done that for all repositories I was free to continue:

# install BerkleyDB
wget http://downloads.sleepycat.com/db-4.3.27.tar.gz
tar zxvf db-4.3.27.tar.gz
cd db-4.3.27/build_unix/
../dist/configure
sudo make install


# grab and unpack the Subversion source
wget http://subversion.tigris.org/tarballs/subversion-1.1.4.tar.gz
tar xzvf subversion-1.1.4.tar.gz 
cd subversion-1.1.4


# remove old copies of libraries as per INSTALL
sudo rm -f /usr/local/lib/libsvn*
sudo rm -f /usr/local/lib/libapr*
sudo rm -f /usr/local/lib/libexpat*
sudo rm -f /usr/local/lib/libneon*


# build
sh ./autogen.sh
./configure --with-berkeley-db=/usr/local/BerkeleyDB.4.3
make
sudo make install
svn --version

Now I was ready to recreate my old repositories and insert the contents back in:

# go back to where you put the dumps
cd dumps


# create a new empty repository
sudo svnadmin create ~/Developer/svnrep/Synergy
sudo chown -R svn:svn ~/Developer/svnrep/Synergy


# restore its contents from the dump
sudo svnadmin load ~/Developer/svnrep/Synergy < \
                   Synergy.dump
sudo chown -R svn:svn ~/Developer/svnrep/Synergy


# enjoy the fruits of your labour
sudo /usr/local/apache2/bin/apachectl start

And that's all for, until next time... (looks like we can expect a 1.2 release soon).