Amess was reportedly stabbed 'multiple times' by a male attacker.
Southend West MP Amess was speaking to constituents at Belfair's Methodist Church.
A statement from Essex Police reads: "A man has been arrested on suspicion murder after a man was stabbed in Leigh-on-Sea.
"We were called to an address in Eastwood Road North shortly after 12.05pm today (Friday 15 October).
"We attended and found a man injured.
"He was treated by emergency services but, sadly, died at the scene.
"A 25 year-old man was quickly arrested after officers arrived at the scene on suspicion of murder and a knife was recovered.
"He is currently in custody.
"We are not looking for anyone else in connection with this incident."
Earlier on, police had appealed for anyone with information or evidence regarding the horrific incident to come forward.
Their statement said: "We were called to reports of a stabbing in Eastwood Road North, #LeighonSea shortly after 12.05pm today (Friday 15 October).
"Officers were on scene shortly after, arrested a man and recovered a knife. He is currently in custody.
"We are not looking for anyone else in connection with the incident and do not believe there is an ongoing threat to the wider public.
"We want to thank the public who alerted us to the incident so quickly.
"We need anyone who saw anything or has CCTV, dash cam or doorbell footage to contact us. If you have any information you can submit a report online at https://www.essex.police.uk or use the 'Live Chat' button to speak to an online operator between 7am-11pm.
"Please quote incident 445 of 15 October. You can also call us on 101 or contact Crimestoppers anonymously on 0800 555 111."
Tributes from across the political spectrum have flooded in since the news was broken.
Awful, tragic news about David. A dedicated, thoughtful man and a true Parliamentarian, who lost his life while serving the constituents who he worked relentlessly for throughout his career.
My thoughts are with his family and friends at this time.
- Rt Hon Grant Shapps MP (@grantshapps) October 15, 2021
Brendan Cox, the husband of murdered MP Jo Cox, said: "My thoughts and love are with David's family. They are all that matter now.
"This brings everything back. The pain, the loss, but also how much love the public gave us following the loss of Jo. I hope we can do the same for David now."
Education Secretary Nadhim Zahawi said: "Rest In Peace Sir David. You were a champion for animal welfare, the less fortunate, and the people of Southend West. You will be missed by many."