The 31-year-old had seemed set to join the foreign legion in the Top 14 after holding talks with the Parisians.
However with the financial issues at Stade scuppering the deal, Exeter have taken the chance to sign Short on a 12-month deal.
Short will offer a wealth of experience to the Exeter pack, having made 145 appearances for Bath during six years at The Rec.
And Chiefs coach Rob Baxter is delighted with his new addition.
He said: "We had a look at him before he decided to go to Stade Francais.
"We met him a while ago but it looked like that had gone cold for us. With developments at Stade things changed and when he became available again we met him again and had another chat and watched more games because he came back fit towards the end of the season, including playing against us.
"He didn't play much last season, but we feel he can add something to the squad.
"He's the type of player I was looking to bring in to the front five. He's a bit bigger and sometimes that element of bulk is what you need in certain games and at certain times of the year.
"He's also a good line-out forward and good at restarts, which sometimes you don't see as the job of the bigger second rows.
"He's capable of playing in the back row and that's a good fit for us because we ask a lot of our front five, we expect them to work very hard and get through a lot of work.
"He's a bit older and the benefit of that is that he brings a lot of experience to a younger pack. He ticks a lot of boxes for us so it was an easy decision to go for him once he became available again."
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