Australian comic Hannah Gadsby has flown a little under the radar during her comedy career, creating perfectly serviceable work that has been received with general warmth. But nothing on her CV could have prepared anyone for the drama and poignancy that's conjured up in Nanette, the show that she insists is her farewell to stand-up comedy. Already this year's winner of the Barry Award in Melbourne, the smart money is on her to scoop the Edinburgh Comedy Award this Saturday.

Even if the gong ends up in someone else's lap, this show will be prove to be a hugely memorable epitaph for Gadsby's stand-up life. Nanette was initially meant to be about an interesting barista she met, but when it turned out that Nanette was neither special nor interesting after all, Gadsby changed tack, dug deep inside herself, and produced one of the most powerful stand-up hours the Fringe has witnessed for many a long year. The curtain is descending on her comedy career because the artform is no longer supplying her with the soothing qualities which it can create in allowing comics to turn their anxieties and traumas into laughter for strangers. Indeed, her ability to seal up the psychological pain she's carried since childhood, and filter it through comedy has, if anything, made matters worse.

Gadsby admits to being permanently furious ever since the debate over same-sex marriage in Australia raised the usual bigoted nonsense about warning of the dangers of gay people bringing up children. For as a child in a resolutely heterosexual environment, Gadsby suffered abuse. Further traumatic incidents in her life are rerun here to staggering effect with a hushed and tearful crowd listening to her castigate those who peddle homophobia while also railing against the comedy industry which insists that pain can be laughed into submission. For Gadsby, this is simply not true, and so she's walking away.

The first half of Nanette is already her best stand-up to date as she makes a foray into re-evaluating comedy's purpose: she got into the game with Bill Cosby acting as an avuncular inspiration. Later she wears her art history background on her sleeve by taking apart the notion of creating a split between the artist and the human being, focusing hard on the misogyny and paedophiliac actions of Pablo Picasso. But anyway, all of that unpleasant stuff can be swept under the carpet because, hey, he created cubism.

Soon, we entered the dark period of the hour, as Gadsby plays with the comedy convention of creating tension before alleviating it with a punchline. She is in no mood to play that game anymore; instead, she constructs tension and insists that we sit and let it wash over us, as she details the horrific violence inflicted upon her as a child, as a teenager and as a young woman.

The power of Nanette is undeniable and this show will last in the mind long after the final discarded Fringe flyer has been swept away. Hannah Gadsby's career has perhaps always been pointing to this very moment, when it's hoped that some catharsis has been achieved by getting this anger out.

Assembly George Square Studios, until 27 Aug, 5.30pm, £11–£12 (£10–£11).