Put the measurement water, sugars and cocoa powder in a saucepan and mix together. Heat gently, stirring until the sugar has dissolved. Increase the heat to bring the mixture to a boil, then reduce to a simmer for 8 minutes.
Remove the pan from the heat and stir in the chocolate, vanilla extract and espresso powder until thoroughly dissolved. Pour into a bowl and cool over ice or leave to cool and chill.
Freeze in an ice-cream machine according to the manufacturer's instructions. Serve immediately or transfer to a chilled plastic freezerproof container and store in the freezer for up to 1 month. If you are using the sorbet straight from the freezer, transfer to the refrigerator 20 minutes before serving to soften slightly.