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In this paper, we characterize the equilibrium of a duopoly market which exhibits advertising spillovers. We show that the relevant distinction is not between purely positive and purely negative advertising spillovers, but between strongly positive and weakly positive or negative spillovers. Moreover, we show that, according to the intensity of advertising spillovers and the cost of the advertising agency, either the two advertisers are active on the market, or one of the two monopolizes the advertising market. In the latter case, under certain conditions, the active advertiser may even push their rival out of the product market. Thus, advertising and the resulting spillovers may be the source of a monopoly situation, leading to an asymmetric equilibrium from an initially symmetric configuration.