The 27-year old initially spent several seasons at the Ospreys, but a split-kneecap almost ended a promising career.
Bishop spent three years battling to remain in the game and successfully got back into top-flight rugby with a spell at Sale in the Aviva Premiership last season.
He spent the early part of this season at Rodney Parade and played a part in the Dragons' pre-season friendlies, but the stint didn't bring the reward of a contract and he has now been registered in Edinburgh's Heineken Cup squad.
"The move has the potential to be a real win-win situation with me getting the chance to show what I can do and Edinburgh getting a bit more strength in depth in the backs", said Bishop.
"I'm happy to play wherever I'm picked, and will be pushing hard in training to earn selection.
"I've played against Edinburgh a few times and I'm well aware of the talent that exists in the squad. I'm excited about the calibre of player I'll be working alongside and I really can't wait to get started."
Moffat was delighted to sign Bishop, especially as it gives him more options in a backline shorn of players through injury and potential Scotland call ups.
"David has been involved with a number of clubs at a high level, and we're happy to have that know-how on board," said Moffat
"Before making him an offer, I spoke to a number of people who've worked with him, and everything they had to say was positive. He's here to put his hand up, and I'm looking forward to him challenging the other guys in training.
"With a few guys away at the Commonwealth Games, and a few others like Nick De Luca, Andrew Easson and Simon Webster still coming back from injuries, David will give us an added bit of depth. I'm really pleased we've got him."
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