The first step is to dump all the information of your current repository into a file. The following command will dump the repository named reponame (located at /svn_old_path) into the file named svndump.
sudo svnadmin dump /svn_old_path/reponame > svndump
Then, copy the svndump file to the new server. I am assuming that subversion is already installed on the new server, if not, this is the guide that I followed:
I setup subversion to access via WebDAV protocol with SSL encryption (https).
On the new server, I will place all subversions repositories under /home/svn. First, we need to create a new repository:
sudo svnadmin create reponame
Now, use the load command to restore all information from the dump file into the new repository:
sudo svnadmin load reponame < svndump
Since I am using subversion access through https, I need to set the proper permissions on the files. In my case, the apache user is www-data and I created a group called subversion.
sudo chown www-data reponame -R
sudo chgrp subversion reponame -R
Finally, change the subversion configuration in Apache and add the new repository. In my case, the configuration file is /etc/apache2/mods-available/dav_svn.conf, and the changes are:
AuthName "Subversion repository"
Then restart apache:
sudo /etc/init.d/apache2 restart