Gelhardt, 24, returns to the club on a permanent four-year deal with the option of a further year, having spent the past 18 months on loan at Hull across two spells, scoring 14 goals in the Championship last season including the tie-clinching second goal in the play-off second-leg win over Millwall that sealed promotion.
Gourna-Douath joins on a five-year contract after four seasons at Salzburg, becoming Hull's 11th signing of the summer as they prepare for a return to the Premier League.
"I'm over the moon," Gelhardt said. "It's been a great 18 months playing here — I've loved every minute of it. Since the season ended, I knew exactly where I wanted to be. Thankfully, it's happened. As soon as I joined the club, I connected with the fans, the staff, all the players. I'm happy to be here on a permanent now, rather than a loan.
"Last season was amazing for everyone involved. A lot of people wrote us off and didn't believe we could do it. We knew the quality we had and there was such a togetherness at the club. I was delighted with the way it happened at Wembley, in the last minute.
"Everyone is aware of how much love I have for Hull. Once the owner called, it was always going to be Hull. Now we're in the Premier League, everyone is excited to play against the best and test ourselves. Thanks to the fans for their support and the love you have shown me. Here's to another four years!"
Gelhardt began his career at Wigan before joining Leeds United, where he scored three goals in 57 appearances around loan spells at Sunderland and Hull City.
Gourna-Douath started out at Saint-Étienne, making 65 appearances there before his move to Salzburg in 2022, where he went on to make 95 appearances across four seasons before spending the second half of last season on loan at Le Havre, featuring 15 times in Ligue 1.
"I'm really happy because it was such a long journey from the first phone call until I signed," Gourna-Douath said. "I'm so excited to get started. I know Hull City is a big family club. Playing in front of our fans will be fun. The Premier League is the best league in the world.
"I love fighting to win the ball back and having the ball at my feet to make the team play. I love to play for the badge and for the fans. I got a chance to play in the Champions League, Serie A and Ligue 1. I've learned a lot and I'm really happy to put this experience on the pitch in the Premier League. The Premier League is the dream of all footballers, so I am excited to play."