Mack, The 1' foot bed will work but the roots will be constrained somewhat and will just grow in a linear way instead of going out in all directions. The feeder roots are about 1" below the surface and can go out in all directions up to 2' in gathering nutrients. Dahlias can grow well while being crowded so it should work but 2' wide would be better.
The ideal raised bed for dahlias is a width of four feet and as long as you would like. This allows tending of the plants from both sides of the bed without walking in the beds. 10" or 12" wide boards are best. 8" boards will work but most articles I have read recommend the higher boards.
I grow in raised beds 8'x 12' with some 8" and some 10" boards. If I had to do it over I would have 4' wide beds.
Check out the Growing Guide on the DS of GA website if you haven't already, under the menu item Dahlias That Grow Well in the South.

Here is a link that might be useful: Dahlia Society of Georgia