The moduli space of (1, 3)-polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y, respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the L-function of their common third e A tale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique normalized cusp form of weight 4 for the group Gamma(1)(6). By Tate's conjecture, this should imply that Y, the fibred square of the universal elliptic curve S-1(6), and Verrill's rigid Calabi-Yau ZA(3), which all have the same L-function, are in correspondence over Q. We show that this is indeed the case by giving explicit maps.