Cardboard does wonders. Put soil on top to keep it flying away. It will compost in situ after the plants under it die.

You can also plastic tarp the area. I worked a garden in spring that was 100% dirt under a black tarp. A worker that was there the year before was blown away. She said the whole field was green with various plants. 

 I like the tarp idea. It’s not a huge garden, probably about 10x10. I could tarp it until spring and then till in ash from our wood stove. Then lay cardboard, etc. 

 You wouldn't need cardboard at that point, but it wouldn't hurt. It sounds like a solid plan. Be sure to take pics! 😀