Godley shared the news on social media, tweeting a picture of herself in a hospital bed.
The 60-year-old Glasweigan said she has “done nothing but cry and cry and cry” over the last day but pledged to use science and technology to cope on the next stage of the journey.
Thousands of people have responded to the original tweet sending best wishes.
Godley was due to appear at the Festival Theatre in Edinburgh on Friday and The Brunton in Musselburgh on Saturday.
She said she arranged an appointment at hospital on Thursday after suffering from a bloated feeling in her stomach.
In a video message posted on Twitter she said: “I’m really sorry I can’t make the Musselburgh and Edinburgh shows this weekend, it’s the very last dates of the tour and the tour has been so brilliant and so very well attended and successful but unfortunately I was doing the tour while I unbeknownst to me had ovarian cancer.
“I had it checked out yesterday, that’s what I’m doing in this very beautiful but very stormy corner room in hospital in Glasgow.”
She added: “Onto the next step of my journey and what is that journey? I don’t know if I’m going to be one of these people who battle bravely, who fight every step, I don’t know.
“That doesn’t feel how I’ve been the last day, I’ve done nothing but cry and cry and cry but I’m going to use science, I’m going to use technology and I’m going to try and cope with this next step on the journey.”
She urged anyone with symptoms to get them checked out and thanked the NHS for being “unbelievably brilliant.”
Godley said: “If you have any symptoms like I had, which was I felt as though I was full before I was eating a meal, I felt as though I had no real energy and there was a big bloatingness inside my stomach, I got it checked out and it’s ovarian cancer.
“So onwards and upwards, I want everybody to know that I really appreciate them for being behind me for so long and we will see what happens next.”
The comedian found viral fame with her dubbed pastiches of Scotland First Minister Nicola Sturgeon’s coronavirus news briefings during the pandemic, which quickly led to the catchphrase “Frank Get The Door”.
She featured in Scottish Government coronavirus adverts but they were pulled after offensive tweets by her came to light following an investigation by the Daily Beast website, with the most recent being from 2017.
The news of her offensive tweets broke soon after she featured in Covid-19 adverts, for which she was paid £12,000.
The adverts were pulled after offensive historical tweets by her came to light.
Following the revelation, Godley profusely apologised for the tweets and donated the fee she was paid by the Scottish Government to charity.