Preheat oven to 350°F. Butter and flour8-inch square baking pan. Using electricmixer, cream butter and 11/2 cups powderedsugar in large bowl. Beat in eggs 1 at atime. Beat in milk, then flour.
Transferbatter to prepared pan; smooth top.
Bakecake until tester inserted into center comesout clean, about 35 minutes.
Meanwhile, finely grate enough limepeel to measure 1 tablespoon. Halve limes;squeeze enough juice to measure 1/4 cup.
Mix peel, juice, and 1/4 cup sugar in smallbowl. Set lime syrup aside.
Using skewer, poke holes all overbaked cake. Spoon half of lime syrup (about3 tablespoons) over hot cake. Cool.
Whisk 1 cup powdered sugar intoremaining lime syrup; drizzle over cake.