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We prove that the hyperelliptic Torelli group is generated by Dehn twistsabout separating curves that are preserved by the hyperelliptic involution.This verifies a conjecture of Hain. The hyperelliptic Torelli group can beidentified with the kernel of the Burau representation evaluated at t=-1 andalso the fundamental group of the branch locus of the period mapping, and so weobtain analogous generating sets for those. One application is that eachcomponent in Torelli space of the locus of hyperelliptic curves becomes simplyconnected when curves of compact type are added.

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=-1