The Hello singer was due to appear in Phoenix, Arizona, for a second night as part of her sell-out world tour.
But the star posted a video on social media explaining she had "pushed too hard" on her first night at the city's Talking Stick Resort Arena.
Adele said she was "so sorry", adding: "It's the last thing I want to do but I have to look after myself."
In the video, posted on Instagram and Twitter, the 28-year-old said: "I'm really, really sorry. I've had a cold for a couple of days and I've been pushing through it.
"I did my show last night and I loved it but I pushed too hard and my cold has gotten the better of me."
Speaking with a croaky voice and appearing emotional, she added: "As you can hear, even if I did the show it wouldn't be very good."
Adele is due to perform two concerts in Los Angeles this weekend and said she would reschedule the cancelled gig, which was due to take place on Wednesday night.
"I'm so sorry to let you down," she said. "I know lots of people will be flying in from all over. It's the last thing I want to do but I have to look after myself because I've got so many more shows to do.
"I will reschedule this show, I promise you. And I will come back and I'll smash it for you."
She said she had been "trying to sing all morning" but added: "Nothing's really happening."
The singer is due to complete her tour in Mexico City in November.
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Adele has apologised to fans after cancelling a concert in America because she is suffering from a cold.