What promise did Dr. Jekyll make to Utterson?

Are you sure you aren't looking for the promise that Utterson made to Jekyll?

Utterson, I wish you to promise me that you will bear with him and get his rights for him. I think you would, if you knew all; and it would be a weight off my mind if you would promise."
"I can't pretend that I shall ever like him," said the lawyer.
"I don't ask that," pleaded Jekyll, laying his hand upon the other's arm; "I only ask for justice; I only ask you to help him for my sake, when I am no longer here."
Utterson heaved an irrepressible sigh. "Well," said he, "I promise."

Jekyll does swear to Utterson that he is "done with him in this world," but there is no promise involved.

"Utterson, I swear to God," cried the doctor, "I swear to God I will never set eyes on him again. I bind my honour to you that I am done with him in this world. It is all at an end. And indeed he does not want my help; you do not know him as I do; he is safe, he is quite safe; mark my words, he will never more be heard of."