While many centrality measures for complex networks have been proposed, relatively few have been developed specifically for weighted, directed (WD) networks. Here we propose a centrality measure for spread (of information, pathogens, etc.) through WD networks based on the independent cascade model (ICM). While deriving exact results for the ICM requires Monte Carlo simulations, we show that our centrality measure (Viral Centrality) provides excellent approximation to ICM results for networks in which the weighted strength of cycles is not too large. We show this can be quantified with the leading eigenvalue of the weighted adjacency matrix, and we show that Viral Centrality outperforms other common centrality measures in both simulated and empirical WD networks.