Preheat the oven to 350 degrees F. Butter a 6-cup loaf pan and line it with parchment or waxed paper. In a medium bowl, combine the flour, baking powder, and salt.
In a mixer fitted with a whisk attachment (or using a hand mixer), cream the butter. Add 1 cup of the sugar and mix. With the mixer running at low speed, add the eggs one at a time. Add the vanilla.
Working in alternating batches, and mixing after each addition, add the dry ingredients and 1/4 cup of the lemon juice to the butter mixture. Mix until just smooth.
Pour into the prepared pan and bake until raised in the center and a tester inserted into the center comes out dry and almost clean (a few crumbs are OK), 65 to 75 minutes.
Meanwhile, make the glaze: In a small bowl, stir together the remaining 1/3 cup sugar and the remaining 1/3 cup lemon juice until the sugar is dissolved.
When the cake is done, let cool in the pan 15 minutes (it will still be warm). Run a knife around the sides of the pan. Set a wire rack on a sheet pan with sides (to catch the glaze) and turn the cake out onto the rack. Peel off the waxed paper.
Using a turkey baster or pastry brush, spread glaze all over the top and sides of the cake and let soak in. Repeat until the entire glaze is used up, including any glaze that has dripped through onto the sheet pan. Let cool at room temperature or, wrapped in plastic wrap, in the refrigerator (Well wrapped, the cake will last up to a week). Serve at room temperature, in thin slices.