Standing next to her mother and brother, she pledged to "faithfully and impartially" discharge her duties.
The public ceremony followed a private one minutes earlier in which she promised to support and defend the constitution.
Sonia Sotomayor is President Barack Obama's first nominee to the nine-member court.
Her appointment - as a successor to liberal Justice David Souter, who retired - is not expected to change its ideological balance.
Ms Sotomayor was born to poor Puerto Rican parents on a New York public housing project, rising to become a respected judicial scholar and judge.
Some Republicans had been critical of her record of speeches - and some rulings - saying they had revealed that she allowed her opinion to affect her decisions.
During her confirmation hearing before the Senate Judiciary Committee, she was asked repeatedly about a speech in which she had remarked that "a wise Latina woman with the richness of her experiences would, more often than not, reach a better conclusion" than a white male judge.
Critics said the remark could have been perceived as racist, but Ms Sotomayor maintained the comments had been an attempted "play on words" that "fell flat".
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