The Center Conjecture for Artin groups proposes that the center of any infinite type Artin group is trivial. This is known to hold for a wide class of Artin groups, but is not known in general. In this talk we will prove that the Center Conjecture passes to the Artin groups whose defining graphs are cones, if the conjecture holds for the Artin group defined on the set of the cone points. In particular, it holds for every Artin group whose defining graph has exactly one cone point. This is joint work with Kasia Jankiewicz.