Combine blackberries, sugar and lemon rind and juice in saucepan. Heat over medium-high heat 5 to 7 minutes, until berries have released their liquid and can be easily broken apart with a spoon. Cool completely.
Heat oven to 375°F. Coat 15 x 10 x 1-1/2-inch jelly-roll pan with nonstick cooking spray. Line with waxed paper; coat paper with spray. Dust clean kitchen cloth with 2 tablespoons of the confectioners sugar.
Whisk flour, baking powder and salt in small bowl. In medium-size bowl, with electric mixer on medium speed, beat eggs for about 5 minutes until thick and lemon colored. Gradually add granulated sugar until well blended. Beat in lemon rind and juice and vanilla. Gently fold in flour mixture until well combined. Scrape batter into prepared pan, spreading evenly.
Bake in 375°F oven for 12 to 15 minutes or until top springs back when lightly touched.
Immediately invert cake onto cloth dusted with confectioners sugar. Remove paper; roll up cloth and cake together from a long side. Cool completely on rack. Unroll. Spread evenly with filling; reroll from long side. Dust with remaining 2 tablespoons confectioners sugar.