Biggins, actor and I'm A Celebrity winner, told the story to Kate Thornton on her White Wine Question Time podcast, saying Dench 'literally fell on the floor' when the Hollywood star emerged from the loos in a dressing room.
He said he was working with Dame Judi on London Assurance in the West End during the 1970s, and because he was the youngest in the theatre company, it was his job to collect money when people were leaving.
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He discussed with Donald Sinden, the head of the company, about presenting gifts to Dame Judi at the end of the show.
He explained: "I went to Donald and I said: 'Donald tonight, Judi's leaving as you know. Will you present her with her present?'
"He said: 'I'd love to, but I've got someone coming to see the show today, who might be slightly better than me. I'm going to tell you who it is. But don't tell anybody.'
"So he told me. I was beside myself! The end of the show came and we all went to Donald's dressing room.
"Judy was there and Donald made a speech, he said: 'Judy, sadly, you're leaving, but we collected for something for you and we couldn't just get any old person to present you with this.'
"And he said: 'So we've got Princess Grace of Monaco', and in she came from his lavatory!
"She came in and — oh my god — it was unbelievable. She was so beautiful. She was so gracious.
"She was funny and Judi Dench fell on the floor. She literally fell on the floor! It was just wonderful."
Biggins explained that he had also introduced Jeremy Irons to his second wife, Sinead Cusack, through the show having tagged along on Irons's honeymoon with his first wife.
Thornton asked him how it happened that he ended up going along on their honeymoon with them. He said: "I was so egotistical. I just wanted to go on a holiday!
"We went to Spain and we were in a Citroen 2CV and I sat in the back. I don't think I'd ever been abroad before, and I didn't know about sun rays.
"When I arrived in Spain with the roof down on this car, I was completely burnt form head to toe. And they had to spend this first week of their honeymoon putting lotion [on me] as I lay in a bed and did nothing, they had to bring food to me!"
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