Wilde was a student at the college and gave the ring to William Ward in 1876.
Prof Sir David Clary, president of Magdalen, said he was "delighted" the ring had come home.
Mr Brand, who was unable to attend the presentation, procured it with the help of commodity broker George Crump, who had "knowledge of the London criminal underworld", The Telegraph reported.
The art detective previously said there were "very strong indications" the ring was taken in the Hatton Garden jewellery raid in 2015.
He said: "We heard these rumours a few weeks after the robbery. We heard very strong rumours it was linked to this theft.
"It ended up in the hands of a person who nearly had a heart attack when they realised it was Oscar Wilde's ring."
Mr Crump explained to the BBC that he "got the word out" about the ring among his contacts.
"I didn't think it would come back - it was impossible - and basically, lo and behold, A went to B, to C, to D, down the line," he said.
"It was recovered in Hatton Garden which was an unbelievable moment.
"I was absolutely astonished. A small thing like that that could have been chucked away and lost forever. A needle in a haystack."
College bursar Mark Blandford-Baker said: "We'd long given it up as having been melted down".
Sir David said: "We thought it had disappeared, so it was a surprise when we heard it had been found.
"We could hardly believe it.... We're very grateful to everyone who helped bring the ring back."