We study the representation theory of a conformal net $A$ on the circle from a K-theoretical point of view using its universal C*-algebra $C^*(A)$. We prove that if $A$ satisfies the split property then, for every representation $\pi$ of $A$ with finite statistical dimension, $\pi(C^*(A))$ is weakly closed and hence a finite direct sum of type $I_\infty$ factors. We define the more manageable locally normal universal C*-algebra $C^*_{ln}(A)$ as the quotient of $C^*(A)$ by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if $A$ is completely rational with n sectors, then $C^*_{ln}(A)$ is a direct sum of n type $I_\infty$ factors. Its ideal $K_A$ of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of $C^*(A)$ with finite statistical dimension act on $K_A$, giving rise to an action of the fusion semiring of DHR sectors on $K_0(K_A)$. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.