Normally when faced with eating two foods where one takes a "mizonos" blessing for grain products, and another takes an all-encompassing "shehakol" blessing, we try to eat the mizonos food first. How do you do this with an ice cream cone where you can't get to the cone without first eating the ice cream which is shehakol?
OU.org says the cone does not need a brocho; it is subordinate to the ice cream.
Even if ice cream is eaten in a cone, only Shehakol is necessary, since the cone is eaten only because of the ice cream and is clearly subordinate to it. Its purpose is actually not so much to be eaten as to hold the ice cream and to prevent the hands from becoming soiled (see Shulchan Aruch, Orach Chaim 212b; Mishnah Berurah 212:5; Kaf Hachaim 15). However, if you specifically use a sugar cone because you enjoy eating it, Mezonot must also be said before eating the cone.
For the enjoyed sugar cone, I suggest biting a piece off the bottom after saying the brocho.