We have considered a hot neutron star with a quark core, a mixed phase of quark-hadron matter, and a hadronic matter crust and have determined the equation of state of the hadronic phase and the quark phase. We have then found the equation of state of the mixed phase under the Gibbs conditions. Finally, we have computed the structure of a hot neutron star with a quark core and compared our results with those of the neutron star without a quark core. For the quark matter calculations, we have used the MIT bag model in which the total energy of the system is considered as the kinetic energy of the particles plus a bag constant. For the hadronic matter calculations, we have used the lowest order constrained variational formalism. Our calculations show that the results for the maximum gravitational mass of a hot neutron star with a quark core are substantially different from those of a neutron star without the quark core.