But for this to work, you need filesystem-access to the repository. And once you have that (and provided the repository is in FSFS format), you can just copy the repository to its new location (if it's in BDB format, dump/load is strongly recommended).
If you do not have filesystem access, you would have to ask your repository provider to provide the dump for you (and make them delete their repository - and hope they comply)
If you want to move the repository and keep history, you'll probably need filesystem access on both hosts. The simplest solution, if your backend is FSFS (the default on recent versions), is to make a filesystem copy of the entire repository folder.
If you have a Berkley DB backend, if you're not sure of what your backend is, or if you're changing SVN version numbers, you're going to want to use svnadmin to dump your old repository and load it into your new repository. Using svnadmin dump will give you a single file backup that you can copy to the new system. Then you can create the new (empty) repository and use svnadmin load, which will essentially replay all the commits along with its metadata (author, timestamp, etc).
You can read more about the dump/load process here:
Also, if you do svnadmin load, make sure you use the --force-uuid option, or otherwise people are going to have problems switching to the new repository. Subversion uses a UUID to identify the repository internally, and it won't let you switch a working copy to a different repository.
If you don't have filesystem access, there may be other third party options out there (or you can write something) to help you migrate: essentially you'd have to use the svn log to replay each revision on the new repository, and then fix up the metadata afterwards. You'll need the pre-revprop-change and post-revprop-change hook scripts in place to do this, which sort of assumes filesystem access, so YMMV. Or, if you don't want to keep the history, you can use your working copy to import into the new repository. But hopefully this isn't the case.
Now you can import a dump file e.g. if you are migrating between machines / subversion versions. e.g. if I had created a dump file from the source repository and load it into the new repository as shown below.
wget http://prdownloads.sourceforge.net/rsvndump/rsvndump-0.5.5.tar.gz
tar xvfz rsvndump-0.5.5.tar.gz
cd rsvndump-0.5.5
./configure
make
sudo make install
Basically, there are plenty of ways to accomplish the task. The topic is covered in depth in SVNBook | Migrating Repository Data Elsewhere, so I suggest reading the book's section.
Here is a brief description of your options:
It depends on your environment, but there is a great chance that you can simply copy the repository to the new server and it will work. You have to revise repository hook scripts after copying the repo to ensure that they are working as you expect.
You can use svnadmin dump and svnadmin load commands to, ehm, generate full dump and then load it to another repository on another server. You will need to svnadmin create a new clean repository to load the dump into it. Keep in mind that the approach deals with repository history only and does not move hook scripts and repository configuration files! As well, you must have read filesystem access to the original repository to dump it.
Since Subversion 1.7, svnrdump tool is available. Generally speaking, it mimics svnadmin dump and svnadmin load functionality, but operates remotely. You are not required to have read / write filesystem access to original and target repositories as tool operates remotely like Subversion client, e.g. over HTTPS protocol. So you need to have read access to original repository and read / write to the target one.
Another option is to use svnadmin hotcopy command. The command is mostly used for backup purpose, it creates full copy of the repository including configuration and hook scripts. You can move hotcopied repository to another server then.