Local police did not name Amess but said a man had been arrested “on suspicion (of) murder” after the stabbing in Leigh-on-Sea.
“We are not looking for anyone else in connection with this incident,” said Essex police, adding that the man in custody was aged 25, and a knife was recovered.
The force said officers had earlier responded to “reports of a stabbing” shortly after 12:05 pm (1105 GMT).
British media, including Sky News and the BBC, identified the victim as the 69-year-old politician from Prime Minister Boris Johnson’s ruling Conservative Party.
Transport Secretary Grant Shapps called Amess “a dedicated, thoughtful man and a true Parliamentarian, who lost his life while serving the constituents who he worked relentlessly for throughout his career”.
Amess is the latest British MPs to have been attacked at his weekly constituency “surgery” with voters, which are typically held on Fridays when parliament is not sitting.
Pro-EU lawmaker Labour MP Jo Cox was murdered by a far-right extremist in 2016 in the run-up to the Brexit referendum.
The Jo Cox Foundation, which was set up in her memory, tweeted that it was “horrified to hear the news of the attack” on Amess.
“We are thinking of him, his family and loved ones at this distressing time,” it added.
Cox’s widower Brendan described the stabbing of Amess was “as cowardly as it gets”.
“Attacking our elected representatives is an attack on democracy itself. There is no excuse, no justification.”
Another Labour MP, Stephen Timms, was stabbed multiple times during an event in 2010, but recovered from “potentially life-threatening injuries” and is still an MP.
Timms said he was “appalled” at the latest attack.
In January 2000, Liberal Democrat MP Nigel Jones was injured and his assistant killed by a man wielding a ceremonial sword at a constituency surgery in Cheltenham, western England.
Former Conservative leader Iain Duncan Smith expressed concern about the security risk of the local MP “surgeries”.
“When you’re not in your office, and you’re in a public place… that means essentially, that the levels of security that sometimes you’re advised to take can’t be taken,” he tweeted.
Amess, a lawmaker since 1983, had advertised his upcoming surgery at the Belfairs Methodist Church in the small town of Leigh-on-Sea.
Paul Gardiner, 41, a barber at One Barbers Leigh-On-Sea, told AFP that “armed police and an ambulance” were on the scene.
“There’s a couple of choppers around the place,” he said.
There was no immediate comment from Johnson, who was leading an away-day for senior ministers in the west of England.
But his wife Carrie, a former Conservative head of communications, tweeted: “Absolutely devastating news about Sir David Amess.
“He was hugely kind and good. An enormous animal lover and a true gent. This is so completely unjust. Thoughts are with his wife and their children.”
David Cameron, one of his Tory predecessors in Downing Street, said: “Very alarming and worrying news reports coming from Leigh-on-Sea.
“My thoughts and prayers are with Sir David Amess and his family.”
Opposition Labour leader Keir Starmer called the news “horrific and deeply shocking”.
An eye-witness, identified only as Anthony, told radio station LBC that the scene was “completely and utterly swamped by police, ambulance, armed police.
“I saw someone get taken out the building, put into the back of a police car. Apparently he was stabbed quite a few times.”