We analyze the contact process on random graphs generated according to the preferential attachment scheme as a model for the spread of viruses in the Internet. We show that any virus with a positive rate of spread from a node to its neighbors has a non-vanishing chance of becoming epidemic. Quantitatively, we discover an interesting dichotomy: for a virus with effective spread rate $\lambda$, if the infection starts at a typical vertex then it develops into an epidemic with probability $\lambda^{\Theta(\log(1/\lambda)/\log\log(1/\lambda))}$, but on average the epidemic probability is $\lambda^{\Theta(1)}$.