The payment follows a legal battle by several women who say they were tricked into having relationships with officers sent to spy on political activists.
"He presented himself as Bob Robinson, a long-haired leftwing radical," Jacqui told the Guardian.
"In reality he was a member of a secret police unit, the special demonstration squad, and was embarking on a five-year mission to infiltrate environmental and animal rights groups."
The man, identified by the paper and BBC as Bob Lambert, disappeared from her life when the child was 2 years old, going back to his original wife, children and identity.
Now an academic, he was one of several officers who had undercover roles as part of the now-defunct special demonstration squad between 1968 and 2008.
He was confronted in 2012 at a conference by members of London Greenpeace, who said he had infiltrated their group in the 1980s as it campaigned on nuclear and environment issues.
Police later acknowledged that he had been an undercover officer.
Jacqui told the Guardian that as a 22-year-old she became smitten by the "very charming and charismatic" Robinson, her first love. Their son was born the following year. She said she was a trusted activist so no one ever questioned him, particularly as they had a child together.
Jacqui told the paper she believed their son was his first child but she discovered decades later that he had another family, miles away on the outskirts of London, and already had two young children with his real wife. She said she gave birth on a Sunday, and he spent the rest of the weekend with his wife and children. "I had a spy who was being paid by the Government to spy on me, to the extent that he watched me give birth, so he saw every intimate part of me. He was with me for 14 hours giving birth. How did he report that back?" she was reported as saying.
A Scotland Yard spokesman said the force "unreservedly apologises for any pain and suffering that the relationship" caused.
The Guardian reported that police chiefs have maintained that the undercover operatives were not permitted to have sexual relationships with the people they were spying on but that evidence has emerged showing such sexual relationships were routine and often lasted several years.
Police were still resisting legal claims from more than 10 other women who say they have suffered emotional trauma after discovering that their one-time boyfriends were undercover officers, the paper said.
In August, four former officers in the squad were told they would not face criminal charges for forming sexual relationships with women while undercover. Several other legal claims relating to undercover officers are ongoing.
In Jacqui's words
He watched me give birth. To me, he was watching his first child being born. He was there throughout the labour, and that is something so intimate between a man and a woman, to watch your wife, your girlfriend, your partner give birth to your child ... And I shared that with a ghost, with someone who vaporised ... I had a spy who was being paid by the Government to spy on me to the extent that he watched me give birth. ... How did he report that back? ... What use was that for information purposes? ... He [their son] was 36 hours old before I saw him hold him, but Bob had been with him all that time. He had bonded with him for 36 hours before that.
So this spy watched me give birth, the most intimate thing a man and woman can share. And he would have [taken] that secret to his grave, because in all this time he had not told his second wife anything about us.
AFP