Hassan Mahhmood admitted to a year-long campaign of harassment during which he bombarded her with messages after she had used his taxi through the taxi hailing company Halo.
The 27-year-old's lawyer told Isleworth Crown Court that Mahhmood was Jemima's big fan because of her marriage to his cricketing hero and former Pakistan Cricket captain Imran Khan.
He pleaded not guilty to a more serious charge of stalking the campaigner, which was allowed to lie on file.
"Have spoken to the complainant, she is satisfied with that result and clearly it will save a lot of anxiety and stress to her if this course is taken and she is fully behind this decision," prosecutor Ruxana Ali told the court.
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The court was told that the taxi driver from Waltham Forest in north-east London became obsessed with the 43-year- old socialite, also known by her maiden name of Jemima Goldsmith, after she posed for a "selfie" with him.
He went on to use 18 different mobile phones to call her 1,182 times, sent her 203 text messages and "loads of" WhatsApp messages between June 16 last year and July 18 this year.
The prosecutor said that Mahhmood told Jemima that he "loved her", "wanted to know her" and asked her "why he could not be friends with her".
Mahhmood's lawyer Umar Ali told the court that his client had no previous convictions but the prosecutor insisted he should be sentenced on the basis of his "persistent pestering behaviour".