Terrestrial planets covered globally with thick oceans (termed ocean planets)
in the habitable zone were previously inferred to have extremely hot climates
in most cases. This is because ${\rm H_2O}$ high-pressure (HP) ice on the
seafloor prevents chemical weathering and, thus, removal of atmospheric CO$_2$.
Previous studies, however, ignored melting of the HP ice and horizontal
variation in heat flux from oceanic crusts. Here we examine whether high heat
fluxes near the mid-ocean ridge melts the HP ice and thereby removes
atmospheric ${\rm CO_2}$. We develop integrated climate models of an Earth-size
ocean planet with plate tectonics for different ocean masses, which include the
effects of HP ice melting, seafloor weathering, and the carbonate-silicate
geochemical carbon cycle. We find that the heat flux near the mid-ocean ridge
is high enough to melt the ice, enabling seafloor weathering. In contrast to
the previous theoretical prediction, we show that climates of terrestrial
planets with massive oceans lapse into extremely cold ones (or snowball states)
with CO$_2$-poor atmospheres. Such extremely cold climates are achieved mainly
because the HP ice melting fixes seafloor temperature at the melting
temperature, thereby keeping a high weathering flux regardless of surface
temperature. We estimate that ocean planets with oceans several tens of the
Earth's ocean mass no longer maintain temperate climates. These results suggest
that terrestrial planets with extremely cold climates exist even in the
habitable zone beyond the solar system, given the frequency of water-rich
planets predicted by planet formation theories.