In a blender or small food processor, combine the basil, olive oil, lemon juice, garlic and lemon zest.

Process until finely chopped and relatively smooth, scraping down the sides of the bowl occasionally. (Large, tough basil leaves will probably stay fairly chunky, which is fine.)

Place the mayonnaise in a small bowl and whisk in the basil mixture until thoroughly combined.

Store in the refrigerator, where it will keep for up to a week, though the color will dull after 2 days.