The 23-year-old back will leave Bristol at the end of the season in May.
"I had a number of offers from England, France and Japan, but moving back to Wales was the correct decision at this point in my career," said Morgan.
"I feel I can test myself and develop the skills I need to keep knocking on the door for Wales."
Newport Gwent Dragons were also keen to sign Morgan, who can play at full-back or fly-half, and started against Fiji in last year's World Cup.
He is in his second season at Bristol and has scored 167 points in 28 appearances for the Championship side he joined from Ospreys in May 2014.
Morgan was named Championship player of the season in his debut campaign in England.
Blues head coach Danny Wilson worked with the player at Bristol and with Wales under-20s and expects Morgan to play most of the time at full-back.
"He has the ability to cover other positions but it's primarily full-back that we have recruited him for," said Wilson.
"It dovetails in with what Wales have done with Matthew recently. Matthew on our (artificial) surface (at the Arms Park) will be a real threat with his footwork and his ability to beat defenders."
Morgan played 66 games for the Ospreys but found his game-time at fly-half limited by Dan Biggar and the emerging Sam Davies, prompting his move to Bristol.

A 911 caller said "they were going to blow up the Statue of Liberty", the National Park Service said.
Federal police arrived and when sniffer dogs found "an area of interest" near lockers at the statue's base, the island was evacuated.
After a sweep, the area was declared safe by late afternoon.
The city's fire department said it had been called at 12:57 local time (16:57 GMT) for the investigation of "a suspicious package".
Statue Cruises, which takes tourists to the Statue of Liberty National Monument and Ellis Island, said ferries were postponed after officials found a suspicious package at Liberty Island.
Karen Mason, 68, of Cincinnati, Ohio, said they had just boarded a boat at Ellis Island at 12:40 p.m.
"All of a sudden, they said 'Disembark, disembark, disembark. We have to evacuate the boat,'" she said.
There was no sense of danger, she added. "But we saw helicopters and we knew something was going on," she said.
The statue was given to the US by France in 1886, in recognition of the friendship between the two nations.

Tettey, who scored a "toe-poke" against Manchester United last month, side-footed home to give Norwich a 1-0 win.
"If I score a goal it will be talked about. Today it was nice to do a Yaya Toure," he told BBC Radio Norfolk.
"I was happy, when I made contact with the ball I looked up and it was just going in, it was a great goal."
Victory at Carrow Road against the Saints was Norwich's third win in their last four Premier League matches and left them 15th in the table, six points clear of the relegation zone.
That followed a run of one win in 11 in all competitions - and the upturn in form has coincided with goalkeeper Declan Rudd coming into the first team for the last six matches.
"For the defenders and Declan as well, he's come into the team and got two clean sheets in a row now," said Tettey.
"For the whole team we are really pleased - where we are, if you're letting in goals you are going somewhere you don't want to be.
"And if you win and don't let in goals you are looking forward - that's what we all want."
Matthew Morgan says he turned down offers from around the world to join Cardiff Blues in the pursuit of adding to his five Wales caps.
[NEXT_CONCEPT]
Hundreds of tourists were evacuated from New York's Statue of Liberty and Liberty Island as a precaution due to a bomb threat and suspicious package.
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Norwich City midfielder Alex Tettey has likened his winner against Southampton on Saturday to those scored by Manchester City's Yaya Toure.