A secondary battery is comprised of a positive electrode, a negative electrode formed from a lithium storage material, and a non-aqueous electrolyte. The non-aqueous electrolyte includes a lithium salt, non-aqueous aprotic solvent(s), such as ethylen carbonate, propylene carbonate, dimethyl carbonate, ethymethyl carbonate and diethyl carbonate, and a small percentage of at least one organic additive. The negative electrode may comprise a carbon such as graphite, and the positive electrode may comprise a lithiated metal oxide or phosphate, such as LiCoO₂, LiNiO₂, LiMn₂O₄, LiFePO₄, or mixtures thereof. The organic additives have one or more unsaturated bonds activated with respect to oxidation by electron-pushing alkyl groups. They are in most cases known to be able to undergo polymerization reactions, such as an anodically induced polymerization especially under certain conditions. The additives are oxidized at the cathode at a potential of more than 4.3 V vs. Li/Li⁺. With these additives in amounts of 0.001 to 10%, a passivation layer is formed on the cathodes, and the sensitiveness of the battery against overcharge is reduced. The electrolyte mixtures do not deteriorate the properties of the battery anodes.