take at (someone's) word
To be convinced of another's sincerity and act in accord with his or her statement: We took them at their word that the job would be done on time.
take (someone) at his/her word, to
To believe someone, to regard someone as trustworthy. This locution dates from the sixteenth century, appearing in such sources as Miles Coverdale’s translation of the Bible (1535) and several of Shakespeare’s plays (e.g., “I take thee at thy word,” Romeo and Juliet, 2.2). It also is part of an amusing proverb quoted in David Ferguson’s Scottish Proverbs (1595) and numerous later collections: “Take a man by his word, and a cow by her horne.”