The Bees have paid 12.7m for the Danish playmaker, who has signed a five-year deal with the Premier League club.
The deal is subject to international clearance, but Damsgaard will train his new teammates immediately.
Frank added: "He has been unlucky with injuries in the past year, but we have already seen that he is an extremely skilful player.
"Importantly, he is a player that will fit into our dressing room. He is a great character, he wants to work hard, he wants to work for the team, and he wants to develop."