On a recent visit to Dublin Valerie Hemingway -- the former wife of the writer's son Gregory Hemingway -- declined to go into detail on the "one-night stand" which took place after the opening night of The Hostage in San Francisco which led to the birth of their son Brendan.
But Valerie, who first met Behan at the age of 16 whilst she was staying in a hostel in Enniskerry, Co Wicklow, has revealed the secret in her book on her life with Ernest Hemmingway Running With the Bulls: My Years with the Hemingways.
"Brendan was not a philanderer as far as I know, but he was a tremendous tease and loved to shock. I had often heard him say to pretty young women: 'I would love to take you to bed with me,' -- sometimes with his wife close by. I thought it was all bravado, but I was careful not to put it to the test" she said.
When Valerie returned to Dublin in 1961 from her time in Spain as Hemingway's secretary , she stopped by Anglesea Road to visit the Behans.
"Brendan was there alone and he made a pot of tea, which we drank in the kitchen. Sober, as he was at that moment, he was a most entertaining companion, incorrigibly mischievous and hilariously funny."
Hemingway had very much enjoyed reading Borstal Boy, when it came out in 1958 and Brendan was equally a Hemingway enthusiast.
"I mentioned to Brendan my imminent trip to Cuba to work for the American writer, and he exclaimed, 'Why in the name of Jaysus would you want to work for America's greatest writer when you could work for Ireland's best instead?'" she said.
Behan offered Valerie similar employment to Hemingway, as his general dogsbody, his right-hand person.
The Irish playwright was about to experience great success in 1960 with The Hostage playing on Broadway and Valerie, who was at a loose end, took him up on his offer.
One night after the closing party for The Hostage, at the Mark Hopkins Hotel in San Francisco, Valerie went to bed exhilarated but completely exhausted.
"I was awakened a short while later when Brendan let himself into my room with a key he must have acquired when he made my hotel reservation.
"I learned then that the romantic fantasies he had hinted at during the preceding months were not just attempts at light-hearted flirtation. It was a night that would change my life forever" she writes in Running with Bulls.
By January 1962, Ernest Hemingway was dead and Valerie was convinced that she was expecting Brendan Behan's child.
"In the months after our night together, distracted by Ernest's death and my travels, I ignored the signs, and would not acknowledge to myself that I was pregnant. My friends threatened to tell Brendan if I didn't so one day in late January I took up the phone and dialled Dublin.
"The moment I broke the news to Brendan he was elated. 'I'm coming over,' he said without hesitation. 'I'll be there as soon as I can.'
"Over the next few days, he telephoned me a dozen times. He said he had talked to Beatrice, and if I would agree, they wanted to bring up the child together in Ireland."
But in the end it didn't work out.
Speaking on the Marian Finucane show in November 2009 Valerie described it as an "unfortunate happening, but as it has all transpired my son is a terrific fellow and I was the winner in this I would say. But we did have a couple of rocky years, as it wasn't planned. They didn't teach us family planning at the convents in Ireland back then, we were very innocent."
Valerie named the boy Brendan and decided to raise him herself, after Beatrice Behan reneged on her initial promise to bring the child up in her own household.
Brendan continued to write heartbreaking letters to Valerie about the pain of separation and of missing his little son, but in May 1963 Beatrice was pregnant and a daughter, Blanaid Behan, was born on November 24.
"Brendan telephoned me on the day of his daughter's baptism. When I asked what they had named her, all he replied was, 'I would like to call her Valerie'."
But four months later, on March 20, 1964, Brendan Behan died aged 41 in the Meath Hospital from complications of diabetes and alcohol. "It was very sad for everyone" said Valerie.
Gregory Hemingway, Ernest's son, who Valerie went on to marry, was the only father Brendan Hemingway knew and he adopted him.
"Brendan was four at the time and brought up entirely as a Hemingway. He was completely unaware until he was an adult of the Behan connection. He had no interest then nor since in knowing anything about it nor in discussing it, with me or anyone else" said Valerie.