The Scotland international has agreed a four-year-deal to make the move from Glasgow to the South Coast for the chance to prove himself in the Premier League.
The 26-year-old has impressed Saints boss Mark Hughes with his consistent displays in the Scottish Premiership and now he wants him to boost his options at St Mary's.
Hughes is planning a major overhaul of the squad he inherited last season which he just about managed to steer away from relegation trouble.
And Armstrong's ability to score and create from midfield is something that stood out for Hughes.
The Saints boss said: “Stuart is a player who we have admired for some time, and I am delighted we have been able to bring him here, especially given the competition we faced to sign him.
“We have strong players in the squad and it is important we build on that by adding different qualities.
"It is important we add a greater goal threat to the team as a whole, and to our midfield specifically, and Stuart’s record of scoring and creating makes him an extremely exciting addition in that area.
“He brings with him extensive experience of European football and has developed his game impressively over the past few seasons at a major club in Celtic, where there is a demand not only to perform but also to win. Stuart has done both during his time there.
“Our research tells us he is an excellent young man, intelligent, and a strong team player, and clearly having another really good character in our group is part of what we want here at Southampton.
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“He has shown tremendous drive to improve and has also displayed a great desire to be a part of what we are trying to achieve here, and I believe he will be a real asset to the squad in the years ahead.”
Armstrong said: “Southampton seems a really good fit for me. Watching from afar, I really enjoy the attractive football that they play and I’m very happy to be here.
"Every season, I put demands on myself to create and score goals. I hope going forwards from here that it will be the same this term at St Mary’s.
"I’m looking forward to meeting everyone and I’m very excited to get started. It’s a four-year contract; there will be a lot to learn and I want to hit the ground running."
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