Put a saucer in the freezer, along with three metal spoons.
Rinse the blackberries in a colander under cold running water and drain. Put the berries in a large heavy nonreactive saucepan and toss them with the sugar, lime zest, and lime juice. Cook over low heat, stirring, until the sugar has dissolved and the berries have released a lot of juice, about 10 minutes. Clip a candy thermome