A scenario for the generation of the baryon asymmetry in the early Universe is proposed in which cosmic string loops, predicted by theories where the baryon and/or lepton numbers are gauged symmetries, collapse during the friction-dominated period of string evolution. This provides a mechanism for the departure from thermal equilibrium necessary to have a nonvanishing baryon asymmetry. Examples of models are given where this idea can be implemented. In particular, the model with the gauge symmetry ${\mathrm{SU}\mathrm{}\left(3\right)\mathrm{}}_c\otimes {\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}}_L\otimes {\mathrm{U}\mathrm{}\left(1\right)\mathrm{}}_Y\otimes {\mathrm{U}\mathrm{}\left(1\right)\mathrm{}}_B\otimes {\mathrm{U}\mathrm{}\left(1\right)\mathrm{}}_L$ has the interesting feature where sphaleron processes do not violate the baryon and lepton numbers so that no wash out of any initial baryon asymmetry occurs at the electroweak scale.

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