We consider marriage markets with externalities. We focus on weak externalities, that is, markets in which each agent is primarily concerned about her partner. We formalize and prove the claim that weak externalities are not so significant in the marriage market: in this case, the ω-core and the α-core coincide and are both nonempty. In addition, we show that, if we allow agents to block matchings without changing their mate, the results do not longer hold.