Preheat oven to 180 degrees C. Grease a loaf tin. Sift together the flour, baking powder and salt, set aside.
In a medium bowl cream together the sugar and butter. Beat in the eggs one at a time then stir in the lime zest, nutmeg, cinnamon and allspice. Add the dry ingredients alternately with the milk, stirring after each addition. Pour batter into the prepared pan.
Bake for 75 to 90 minutes in the preheated oven until a toothpick inserted comes out clean. Cool in the pan for 10 minutes before turning out onto a rack to cool completely.