I know that this is not the main issue of this thread, but just to clarify that you are not down and out if your DL got stolen.
And you also don't have to dump your private or rental car in the middle of the trip.
What you do need to do is to go to the police. You will be issued a report of your stolen document(s) (sometimes for a fee), which is good to continue your travels. It is helpful to have the necessary information at hand (DL# etc), which should be also on your rental car's contract.

By the way - driving without your drivers license is something different than driving without a license. The fact that you cannot produce the documentation of your permit does not void your permit nor does it nullify any insurances.
It may incur a small to medium penalty if you cannot produce the license or the police report for a lost or stolen license, but, for example, in Germany the penalty for not having the DL with you is 10 euros, in Austria some 30 euros. So even in the worst case, no end-of-the-world-as-we-know-it scenario.

What you do need to do is to go to the police. You will be issued a report of your stolen document(s) (sometimes for a fee), which is good to continue your travels. It is helpful to have the necessary information at hand (DL# etc), which should be also on your rental car's contract.

Good piece of advice, and it turns out that the rental agency probably had all the information, and I didn't think to ask them if they have a photocopy of my license.