Release: June 1986
aka: Atari 7800 ProSystem
Atari really wanted to get it right this time. They designed a system that would be made for the public. They took polls and surveys of what people wanted, they researched.
The main advancement was 'Marie'. This was a graphics chip with more colours, infinite sprite capabilities, and a cartridge authentication system. This was used to keep third parties from making unauthorized games, so Atari could make the profits. This was the same thing that Nintendo did with its NES.
Atari was learning from the 5200's failure. The Atari 7800 could play Atari 2600 games. An attatchement was later released that could play 5200 games as well. All previous Atari games could now be played on this system, so if you missed out and never bought a 2600 or 5200, you could now buy the new system, play all the old games, and grab yourself some new state-of-the-art software. Or so was the plan.
In 1984, shortly before the system would have been released, Atari was sold by Warner Communications to some guy (Jack Tramiel) who wanted to focus on the computer lineup. The Atari 7800 was put on hold until 1986. The Nintendo Entertainment System had revived the industry, and the Sega Master System was being released. Atari wanted to get in on the money before it was gone, and the 7800 was pulled off the shelf and released. This late release with few games ruined any chance the Atari 7800 had. The system would not make it big.
Learn more about Atari in Scott Cohen's book: Zap.