We define for any algebraic variety X over a field k a complex of abelian groups whose homology groups are called algebraic singular homology groups of X. This definition was suggested by the first author in 1986 and two conjectures were made relating these groups to higher Chow groups (for integral coefficients) and to etale cohomology (for finite coefficients). In this paper we prove the second conjecture. There is an appendix which contains a brief summary of main properties of h- and qfh-topologies which are used in the proof. After this paper was written, the first conjecture was proved by Suslin. See "Higher Chow groups and etale cohomology".