I thought of RAID as the solution, but was frustrated at the complexity of migrating the old but full drive. The solution: LVM.
Large Volume Management (LVM) is a very neat set of tools to concatenate a bunch of hard drives such that they look like one big partition. I followed one of the many LVM + MythTV how-tos out there and was in no time able to get my new drive (400G) to act as LVM, then copy all files from the old drive to the new (took about an hour). Then add the old drive to the LVM and resize the filesystem.
Now I am the proud owner of my first LVM array of drives with 619,972 MB of space!
The nice side note was that I could use the spare 40G drive that was in place of the new 400G one and just place it in a external USB casing. Now I can take a lot of recordings with me.
The last week was branded by my franatic searches for more space on my MythTV server. Hard drive space (250G) was not enough to hold the movies I had been recording. I finaly thought it was time to get a bigger drive, but what to do with the smaller one. My goal was not to just add two drives and mount them at separate places, as this would not work with MythTV. I needed a way of "connecting" both drives such that they would look like one and I did not want to loose my 230G worth of recordings in the process.