The 22-year-old is one of the rising stars of Welsh rugby, having made his debut for his country's second-string back in 2001 and has signed a two-year contract.
He was in Bath's starting line-up for Zurich Premiership ties five times during the 2002/03 season, his opportunities limited by the presence of Wales international Gareth Cooper – another Welshman to have headed home during the off-season.
"Andy is a local boy, with a great attitude and a huge natural talent," said Neath-Swansea Director Mike Cuddy.
"I have no doubt he will be pushing for senior international honours in the not too distant future."
Williams will hook up with former Bath team-mates Elvis Seveali'i and Gavin Thomas, two other recent signings for the as-yet-un-named Welsh regional side who will unveil their new 'brand' on July 23.
The loss of Williams means Bath must accelerate their search for a new scrum-half signing, or find themselves critically short of cover in the coming season.