On the morning following the night when Esther tells Mr. Jarndyce about the true relationship between her and Lady Dedlock, Mr. Jarndyce assures Esther that the secret is safe with him, and that if the situation ever warrants it, he will do Lady Dedlock a good turn, if only for Esther's peace of mind, to protect her--Lady Dedlock--from the machinations of Mr. Tulkinghorn. Esther is grateful and is about to leave when Mr. Jarndyce summons her back. He tells how grateful he is to her and tells Esther there's something he's been meaning to propose to Esther but that if Esther has any doubts, doubts as to the proposal's effect on the current dynamic between Esther and Mr. Jarndyce, i.e. irrevcocably change it for the worse, then to say the word and Mr. Jarndyce will never utter the proposal. Esther has an inkling what the proposal is. She assures Mr. Jarndyce that nothing will ever change her sense of gratitude she feels for Mr. Jarndyce and encourages him to make the proposal. Mr. Jarndyce agrees to do so and asks Esther to wait for it as he will put it in writing.
A week passes when Esther commissions Charley to go to Mr. Jarndyce's office to get the proposal. Charley brings the proposal, and left alone, Esther eventually reads it after a moment of hesitation. The proposal is what she had a feeling of what it would be: Mr. Jarndyce would like Esther to be the mistress of Bleak House. Subsequently, Esther finds the flowers that Mr. Woodcourt had given her before leaving for China/India and burns it to dust.
About a another week passes during which Esther expects Mr. Jarndyce to broach the topic, which he never does. Esther decides to give Mr. Jarndyce a reply in writing but keeps putting it off. Then one day when Mr. Jarndyce and his wards have decided to go out for fresh air, Esther enters Mr. Jarndyce's office in advance of Ada and embraces and kisses him. The answer is yes: Esther will be the mistress of Bleak House.