If a false rumor propagates via Twitter, while the truth propagates between friends in Facebook, which one will prevail? This question captures the essence of the problem we address here. We study the intertwined propagation of two competing "memes" (or viruses, rumors, products etc.) in a composite network. A key novelty is the use of a composite network, which in its simplest model is defined as a single set of nodes with two distinct types of edges interconnecting them. Each meme spreads across the composite network in accordance to an SIS-like propagation model (a flu-like infection-recovery). To study the epidemic behavior of our system, we formulate it as a non-linear dynamic system (NLDS). We develop a metric for each meme that is based on the eigenvalue of an appropriately constructed matrix and argue that this metric plays a key role in determining the "winning" meme. First, we prove that our metric determines the tipping point at which both memes become extinct eventually. Second, we conjecture that the meme with the strongest metric will most likely prevail over the other, and we show evidence of that via simulations in both real and synthetic composite networks. Our work is among the first to study the interplay between two competing memes in composite networks.