What a lot of people do with pot tubers is plant them with the eyes above the level of the potting mix and take cuttings when shoots appear. Do not take any of the tuber with your cuttings, as this will destroy the eyes. Cut just above the base of each shoot. That way the eye remains, and more shoots will emerge. You can usually take three sets of cuttings or more.

Here is a link that might be useful: A good photo of how to take Dahlia cuttings.

Interesting, thanks for the information.
I would have thought that as Dahlia's produce such soft and 'watery' stems, the biggest problem would be from rotting.

Unfortunately I don't have a greenhouse, but I'll nevertheless
give this one a try come the spring.

Rot can be an issue, but they grow from cuttings OK. Here is a bit more on cuttings - this video uses a greenhouse too, but some people raise cuttings on windowsills or heat pads inside during winter, or unheated in spring.

Here is a link that might be useful: Video on Dahlia cuttings.

I'm so glad to read this--does the cuttings then make a new tuber? I'm guessing they do--but since I had never heard of this, I'm going to try it. Do they grow true to what the cutting was from?

The cuttings do not always form a new tuber in their first season, so some people put the cutting in a small pot and plant the pot with the cutting still in it into the garden when they plant out their tubers. In autumn they dig up and store the pot, not disturbing the roots unless it is obvious that tubers have formed. In spring they plant out the root without the pot, and it will grow tubers in that second year.

Here we can overwinter in soil, so it doesn't matter.

Cuttings are true to the parent plant.