The X Factor boyband made history as the first UK group to top the charts with their debut album. The previous highest-placed debut entry from a British group was Spice by the Spice Girls, which entered the Billboard album charts at number six in February 1997.
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Up All Night shifted 176,000 copies in its first week on sale, Nielsen SoundScan figures suggest.
Speaking about their success, singer Harry Styles said: "We simply cannot believe that we are number one in America! It's beyond a dream come true for us.
"We want to thank each and every one of our fans in the US who bought our album, and we would also like to thank the American public for being so supportive of us."
Niall Horan added: "We just found out that we are number one in America and as you can imagine, we are over the moon!
"When we got put together as a group, we couldn't imagine ourselves coming to America, let alone releasing our album here, so for us to be sitting at the top of the US album charts is unbelievable. We are so grateful to everyone who has supported us in the US."
Syco music boss and X Factor judge Simon Cowell offered One Direction his congratulations on their "incredible achievement", adding: "They deserve it. They have the best fans in the world."
'What Makes You Beautiful', which has already been a chart-topping hit in the UK, entered the Billboard Hot 100 at number 28 and became the highest-charting debut entry on the chart for a UK act in the process.
The Brit Award-winning band are in the midst of an extensive promotional tour of the US, having already performed on The Today Show and supported Big Time Rush on tour.
Styles also joked that socialite Kim Kardashian should call him during a recent appearance on Pennsylvania radio station Q102.
One Direction will appear on Saturday Night Live on April 7.
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Watch the music video for 'What Makes You Beautiful' by One Direction below: