Chasing a target of 249 for victory, England were 16 for one off four overs when rain stopped play.
A minimum of 20 overs needed to be bowled in the second innings for a result to be declared under the Duckworth/Lewis method for rain-affected matches.
But the players were still off the field when the umpires abandoned the match at 5:30pm local time (1630 GMT).
On Friday, Sri Lanka had 34.1 overs in the field without any success as England won the second ODI at Edgbaston by 10 wickets to go 1-0 up in the five-match series after Tuesday's tie at Trent Bridge.
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It was a very different story for Hales at the County Ground on Sunday when he fell for nought off the first ball he faced when caught behind chasing a wide delivery from Suranga Lakmal.
- Woakes's fine figures -
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Earlier, Chris Woakes and Liam Plunkett both took three wickets apiece as Sri Lanka were held to 248 for nine after England captain Eoin Morgan won the toss.
Three Sri Lanka batsmen -- Dinesh Chandimal (62), captain
But the trio all fell going for big legside shots, with Sri Lanka's total looking well below par.
England had a wicket just five balls into Sunday's match when Danushka Gunathilaka played on to left-arm quick David Willey when trying to leave the ball.
Kusal Perera fell for nine when he got a top edge off Plunkett and wicket-keeper Jos Buttler held a skyer.
First-change Plunkett had now taken a wicket with his second ball on Sunday after striking with this third delivery on Friday.
Mendis pulled all-rounder Chris Jordan, recalled in place of off-spinner Moeen Ali, for four and six off successive deliveries.
Mathews, like Chandimal troubled by a hamstring problem, drove Plunkett down the ground for six.
Chandimal completed a 65-ball fifty but then, trying to hit Woakes legside, got a thick edge straight to Jordan at third man.
That was the end of a fourth-wicket stand of 80 with Mathews, who later skyed a pull off Jordan to Buttler.
Woakes ended his spell by having Farveez Maharoof caught in the deep and next ball bowling Upul Tharanga (40) middle stump.
The series continues with a day/night match at The Oval in south London on Wednesday.