The 25-year-old central midfielder has put pen to paper on a five-and-a-half-year deal at St. James’ Park, arriving for an undisclosed fee.
Saivet came up through the ranks at Bordeaux, making his first team debut in 2007 at the age of 17.
He went on to make 178 appearances for the Ligue 1 side in all competitions, scoring 23 goals, and featured in both Europa League group games against the Magpies in the 2012/13 campaign.
“I am very happy to be here at Newcastle,” said Saivet.
“I am really proud to be able to play for this Club, and I am really excited at the challenge that lies ahead.
“St. James’ Park is a really impressive and special stadium. In France there is nothing that compares to this, and I can’t wait to get out there with my new teammates and help the team climb the table.”
Newcastle United head coach Steve McClaren said: “Henri is someone the Club have been watching for a while, and he has developed into an excellent player at Bordeaux.
“This is someone with a very good mentality, who is a tough player and a winner. Those are key qualities you need in the Premier League and we are looking forward to seeing him pulling on the Newcastle United shirt.”