This is a joint work with David Lehavi. The Kodaira dimension of the moduli space of principally polarized abelian varieties, A_g, was studied by Freitag, Tai, Mumford, and many others. In particular Mumford finally showed that A_g is of general type for all g>6, and it was shown that A_g is unirational for g<6 (the work of Donagi, Clemens, ...). Thus the borderline case of A_6 remained the only open question. We show that A_6 is of general type by constructing a new geometric divisor on it of slope less than 7. In general our construction defines a rational map from a finite cover of A_g to the moduli space of curves of genus G=(g-1)g!/2+1 with indeterminacy locus of high codimension. This map allows one to pull back geometric divisors from M_G to A_g and thus may be of independent interest.