How to create a 1 block world:

1. Create a new world using the mode Superflat. (More World Options > World Type)
   
2. Under mode, the button Select a Preset should appear, click this.
   
3. Find the text bar located at the top and enter either [1;12;1] or [1;13;1]
   
4. Click Use Preset and Create World
   
5. Find a location and place any block of your choice (Excluding Sand, Gravel, Water or Lava)
   
6. Wait for the rest of the blocks to disappear. If it gets too laggy, exit to the main menu and load the world again.
   

How this works:

When the world is created there will be a layer of flat gravel or sand. When a block is placed Minecraft performs a block update around every block around that, and if that block is changed every block around that and so on. The Sand/Gravel that is next to the placed block will be affected by Minecraft's Gravity Physics and it will fall into the void like dominos. All that will be left will be your 1 block.