The 41-year-old moves to the Liberty Stadium on a contract that will run through to the summer of 2014.
Erik Larsen has been working as Laudrup's right-hand man so far this season, but he will now take on a greater responsibility for international scouting while Alan Curtis will remain as a first-team coach.
Wieghorst has been in charge of the Denmark Under-21 side, and will be given permission to take charge of their two fixtures in March.
During his time in Glasgow, Wieghorst helped Celtic end Rangers' run of nine-straight SPL titles, before being diagnosed with Guillain-Barre syndrome in 2000 and making a remarkable return to resume his playing career.
Wieghorst, who played alongside Laudrup for Denmark, left the Hoops in 2002 to become one of his former team-mate's first signings at Brondby, before enjoying back-to-back Danish Cup triumphs after moving into management with Nordsjaelland.
Wieghorst has been cited as a possible successor to Morten Olsen, when the coach of the senior Denmark side stands down next year, and Laudrup is delighted to have got his man.
He said: "Morten had a good playing career with Dundee and Celtic and the last years of his career he played under me at Brondby and won the league. He went to Nordsjaelland, who have been in the Champions League this season, first as an assistant and then as a manager, winning two cup titles which is huge for a smaller club.
"He has been with the Under-21 national team for the last year and a half. I know Morten, I played with him, I coached him as a player and watched him as a manager. I had some inside information as my son Andreas has played for him.
"I wanted an assistant who had experience of being in charge as a manager, and he has experience of British football."