Austin had to end his basketball career before it had properly taken off after being diagnosed with Marfan syndrome just a week ago.
The rare genetic disorder, which affects the heart, was only discovered in a medical the player took before the Draft.
"It's really been a tough week for me, and it's been really rough," said Austin, according to The AP.
"I've just had a tremendous amount of support from everybody around the world, telling me they're praying for me and everything. I can't thank everybody enough."
The crowd at the Barclay Center rose to its feet when Austin's name was called as Silver explained that the NBA would let Austin fulfil his dream of being picked in the Draft.