Put the blackberries into a 1.5-liter bottle. Zest lime and add zest to bottle. Remove and discard remaining pith and connective membranes from the lime; putting only the remaining pulp into the bottle. Pour tequila into the bottle.
Seal the bottle and store in a cool, dark place for 2 weeks, shaking once daily.
Strain tequila through cheesecloth or coffee filter. Stir simple syrup into the tequila.