In a large saucepan over medium-high heat, saute the rice, onion, dill, and mint for about 5 minutes, or until onion is soft. Pour in 1 quart of broth, reduce heat to low and simmer for another 10 to 15 minutes, or until rice is almost cooked. Stir in 1/2 of lemon juice and remove from heat.
Take one leaf, shiny side down, and place 1 teaspoon of the rice mixture at the bottom (stem) end of the leaf. Fold both sides of the leaf towards the center, roll up from the broad bottom to the top, and place into a 4-quart pot. Repeat with all leaves, leaving no gaps as leaves are placed in pot (to prevent from opening while cooking). Sprinkle with remaining lemon juice and with olive oil.
Pour chicken broth over all to cover grape leaves. Cover pot and simmer for about 1 hour (do not boil, because this will make the stuffing burst out of the leaves). Remove from heat, remove cover and let cool for 1/2 hour. Transfer to serving dish and serve.