The 21-year-old has signed a four-year contract and joins the Saints for a reported club-record fee of £12million.
Southampton manager Nigel Adkins said: “A lot of hard work has gone in from the football club with a lot of negotiations behind the scenes to get this deal done.
“Great credit to everyone because we have a young, exciting talent joining us who is one of the best around.
“The chairman has gone and backed this deal big-time with the financial outlay, and what we have got is one of the most exciting prospects in world football coming to St Mary’s.
“There is a lot of ambition here and we are all delighted that Gaston is going to come and join us.”
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