Preheat oven to 350. Grease 8 1/2 by 4 1/4 by 2 1/2-inch loaf pan. Line the bottom with parchment paper. Grease and flour the pan.
Sift together flour, baking powder, and salt into bowl. In another bowl, whisk together yogurt, 1 cup sugar, eggs, grapefruit zest, and vanilla. Slowly whisk the dry ingredients into the wet ingredients. With a rubber spatula, fold the vegetable oil into the batter, making sure it's all incorporated. Pour batter into the prepared pan and bake for about 50 minutes, or until cake tester placed in center of the loaf comes out clean.
Meanwhile, cook the 1/3 cup grapefruit juice and remaining 1 tablespoon sugar in a small pan until the sugar dissolves and the mixture is clear. Set aside.
When cake is done, allow to cool in the pan for 10 minutes. Remove from pan and carefully place on baking rack over sheet pan. While cake is still warm, pour grapefruit-sugar mixture over cake and allow to soak in. Cool.
For the glaze, combine the confectioners' sugar and grapefruit juice and pour over the cake.