Emma Woodhouse has just attended the wedding of Miss Taylor, her friend and former governess, to Mr Weston. Having introduced them, Emma takes credit for their marriage, and decides that she likes matchmaking. After she returns home to Hartfield with her father, Emma forges ahead with her new interest against the advice of Mr Knightley and tries to match her new friend Harriet Smith to Mr Elton, the local vicar. First, Emma must persuade Harriet to refuse the marriage proposal from Robert Martin, a respectable, educated, and well-spoken young farmer, which Harriet does against her own wishes. But Mr Elton, a social climber, thinks Emma is in love with him and proposes to her. When Emma tells him that she had thought him attached to Harriet, he is outraged. After Emma rejects him, Mr Elton leaves for a stay at Bath and returns with a pretentious, nouveau-riche wife, as Mr Knightley expected. Harriet is heartbroken and Emma feels ashamed about misleading her.
Frank Churchill, Mr Weston's son, arrives for a two-week visit to his father and makes many friends. Mr Knightley suggests to Emma that while Frank is clever and engaging, he is also a shallow character. Jane Fairfax comes home to see her aunt, Miss Bates, and grandmother, Mrs Bates, for a few months, before she must go out on her own as a governess. She is the same age as Emma, but Emma has not been as friendly with her as she might. Emma envies her talent and is annoyed to find all, including Mrs Weston and Mr Knightley, praising Jane. The patronising Mrs Elton takes Jane under her wing and announces that she will find her the ideal governess post before it is wanted. Emma begins to feel some sympathy for Jane's predicament.
Emma decides that Jane and Mr Dixon are mutually attracted, and that is why she has come home. She shares her suspicions with Frank, who met Jane and the Campbells at a vacation spot a year earlier, and he apparently agrees with her. Suspicions are further fueled when a piano, sent by an anonymous benefactor, arrives for Jane. Emma feels herself falling in love with Frank, but it does not last to his second visit. The Eltons treat Harriet badly, culminating with Mr Elton publicly snubbing Harriet at the ball given by the Westons in May. Mr Knightley, who had long refrained from dancing, gallantly steps in to dance with Harriet. The day after the ball, Frank brings Harriet to Hartfield, she having fainted after a rough encounter with local gypsies. Harriet is grateful, and Emma thinks this is love, not gratitude. Meanwhile, Mrs Weston wonders if Mr Knightley has taken a fancy to Jane but Emma dismisses that idea. When Mr Knightley mentions the links he sees between Jane and Frank, Emma denies them, while Frank appears to be courting her instead. He arrives late to the gathering at Donwell in June, while Jane leaves early. Next day at Box Hill, a local beauty spot, Frank and Emma continue to banter together and Emma insults Miss Bates.
When Mr Knightley scolds Emma for the insult to Miss Bates, she is ashamed and tries to atone with a morning visit to Miss Bates, which impresses Mr Knightley. On the visit, Emma learns that Jane had accepted the position of governess from one of Mrs Elton's friends after the outing. Jane now becomes ill, and refuses to see Emma or accept her gifts. Meanwhile, Frank was visiting his aunt, who dies soon after he arrives. Now he and Jane reveal to the Westons that they have been secretly engaged since the fall but Frank knew that his aunt would disapprove. The strain of the secrecy on the conscientious Jane had caused the two to quarrel and Jane ended the engagement. Frank's uncle readily gives his blessing to the match and the engagement becomes public, leaving Emma chagrined to discover that she had been so wrong.
Emma is certain that Frank's engagement will devastate Harriet, but instead Harriet tells her that she loves Mr Knightley, although she knows the match is too unequal. Emma is startled, and realizes that she is the one who wants to marry Mr Knightley. Mr Knightley returns to learn Emma's reaction to the engagement. When she admits her own foolishness, he proposes and she accepts. Now Harriet accepts Robert Martin's second proposal and they are the first couple to marry. Jane and Emma reconcile, and Frank and Jane visit the Westons. Once the period of deep mourning ends, they will marry. Before the end of November, Emma and Mr Knightley are married with the prospect of "perfect happiness".

Question: How did the engagement between Frank and Jane end?
Answer: The engagement between Frank and Jane ended because the strain of the secrecy cause them to quarrel, leading to Jane ending their engagement.
Alternative answer: they'll get married after a mourning period