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Heat the oven to 180C (160C fan)/350F/gas 4, and grease and line a 900g loaf tin.
Put the sugar and grapefruit zest in a large bowl and rub together with your fingertips until the sugar looks a bit like wet sand and it’s smelling fragrant.
Add the eggs and oil, mix to combine, then stir in the grapefruit juice and the yoghurt.
In a separate bowl, combine the flour, baking powder, bicarbonate of soda and salt, then add to the wet ingredients and mix until smooth.
Pour the batter into the prepared tin and bake for 45-50 minutes. It’s ready when a knife or metal skewer is inserted and comes out clean.
Meanwhile, make the syrup. Put the grapefruit juice and sugar in a small saucepan, bring to a simmer for three to four minutes, then take off the heat.
When the cake is out of the oven, prick the top all over with a skewer, then pour the syrup over the cake and leave to seep through and cool.
To make the glaze, add a tablespoon of grapefruit juice at a time to the icing sugar until it’s a thick but pourable consistency.
When the cake is completely cool, spoon the glaze on top, nudging it down the sides, top with some extra zest, then slice and serve.