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Conway reached his double-century with a pull for six and was out without facing another ball, coming up just short as he sprinted back to the non-striker's end for second run.
The South African-born opener was given a standing ovation at the home of cricket, having made the highest score by a batsman on test debut in England. He broke a 125-year-old record held by former England player K.S. Ranjitsinhji, who scored 154 in Manchester in 1896.
Conway is the second New Zealand batsman to score double hundred on Test debut.
New Zealand had resumed on its overnight score of 246-3, with Conway on 136, and the dismissal of Henry Nicholls for 61 triggered the Black Caps' collapse from 288-3.
No. 11 Neil Wagner was 25 not out off 21 balls, adding some valuable runs at the end of the innings.
Ollie Robinson, also making his Test debut, had England's best bowling figures of 4-75.
After stumps on Wednesday, the paceman was close to tears as he apologized for a string of sexist and racist messages that he posted on Twitter from 2012-14 and which resurfaced over social media while the game was being played.
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