It is demonstrated that the inclusion of the geometric phase has an important effect on ultracold chemical reaction rates. The effect appears in rotationally and vibrationally resolved integral cross sections as well as cross sections summed over all product quantum states. The effect arises from interference between scattering amplitudes of two reaction pathways: a direct path and a looping path that encircle the conical intersection between the two lowest adiabatic electronic potential energy surfaces. It is magnified when the two scattering amplitudes have comparable magnitude and they scatter into the same angular region which occurs in the isotropic scattering characteristic of the ultracold regime (s-wave scattering). Results are presented for the O + OH -> H + O-2 reaction for total angular momentum quantum number J = 0-5. Large geometric phase effects occur for collision energies below 0.1 K, but the effect vanishes at higher energies when contributions from different partial waves are included. It is also qualitatively demonstrated that the geometric phase effect can be modulated by applying an external electric field allowing the possibility of quantum control of chemical reactions in the ultracold regime. In this case, the geometric phase plays the role of a "quantum switch" which can turn the reaction "on" or "off".