Position a rack in the center of the oven and heat to 350°F Using a cylinder shaped pastry brush, butter the inside of a 9 to 9 1/2 inch Gugelhupf mold or fluted cake pan. Coat with flour, tapping out the excess.
Beat the butter in a medium bowl using a handheld electric mixer on high speed until smooth, about 1 minute. On low speed, beat in the confectioners’ sugar. Return the speed to high and beat until the mixture is very pale and light in texture, about 3 minutes. One at a time, beat in the eggs, then the lemon zest.
Sift the flour, baking powder, and salt together into a bowl, and stir in the ground poppy seeds. Mix the milk, rum, and vanilla in a glass measuring cup. Stir in half of the flour, then half of the milk, then repeat. Spread evenly in the pan.
Bake until a toothpick inserted in the center of the cake comes out clean, 50 to 60 minutes. Cool on a wire rack for 10 minutes. Invert onto the rack to unmold and cool completely.
MAKE AHEAD: The cake is best the day it is made. Cover leftovers tightly in plastic wrap and store at room temperature. To refresh, heat slices in a toaster oven.
NOTE: Buy poppy seeds in bulk from a purveyor with a steady turnover, such as a central European delicatessen or Middle East or Indian grocer (poppy seeds are used in the cuisines of the East, too). Poppy seeds in tiny jars from the supermarket are likely to be prohibitively expensive and rancid. Store poppy seeds in an airtight container in the refrigerator or freeze them for up to 6 months. Always give poppy seeds a good sniff before using to check for any hint of rancidity.
Poppy seeds are usually ground before use to crack their tough hulls, which releases their flavor and moisture. Use an electric coffee grinder or blender. Grind the seeds in 1/4 cup batches until they look smaller than whole seeds and begin to clump together. A blender takes longer than a grinder, about 1 minute and be sure to stop the blender occasionally to loosen the ground seeds that collect around the blade. Serves 10-12.