How can I play older worlds on newer versions of minecraft?

In the new launcher you can play beta and alpha versions of minecraft. I made a world in alpha 1.2.6 and now I want to take it to beta 1.3.0. But it does not show up in there.

How can I get my world to show up?

Each world should have a save folder, located in the saves folder in your .minecraft directory.

On Windows, it is located in: %AppData%\.minecraft.

All you should need to do is move the old save file into the new Minecraft directory folder. When you open your game, the worlds should appear in the list, and then convert to the new version when opened. After that, they'll work as normal.

On a side note, using the same seed in newer versions of Minecraft won't work the same way, due to different salts applied and variable used to generate the terrain.