If you have a budget of $300, then if you buy new, all you will get is a 30+ piece-of-garbage that will be scrap in less than 2 years.
For $300, you can get a good used bike in the 22-25 lb range.
For sub 20 lb at $300, you'll probably have to remove the seat, the seat post, the bottle cages, and possibly the derailleurs and the brakes to get down to THAT level.....