In a food processor, puree blackberries, water and lemon juice. Strain blackberries, reserving juice and pulp. Discard seeds. Return pureed blackberries to food processor; add sugar and vanilla. Cover and process until smooth.
In a large bowl, combine yogurt and blackberry mixture. Fill cylinder of ice cream freezer two-thirds full; freeze according to the manufacturer’s directions. Refrigerate remaining mixture until ready to freeze. When yogurt is frozen, transfer to a freezer container; freeze for 2-4 hours before serving.