The energy-momentum of a new four-dimensional, charged, spherically symmetric, and nonsingular black hole solution constructed in the context of general relativity coupled to a theory of nonlinear electrodynamics is investigated, whereby the nonlinear mass function is inspired by the probability density function of the continuous logistic distribution. The energy and momentum distributions are calculated by use of the Einstein, Landau-Lifshitz, Weinberg, and Møller energy-momentum complexes. In all these prescriptions, it is found that the energy distribution depends on the mass and the charge of the black hole, an additional parameter coming from the gravitational background considered, and the radial coordinate . Further, the Landau-Lifshitz and Weinberg prescriptions yield the same result for the energy, while, in all the aforesaid prescriptions, all the momenta vanish. We also focus on the study of the limiting behavior of the energy for different values of the radial coordinate, the parameter , and the charge . Finally, it is pointed out that, for and , all the energy-momentum complexes yield the same expression for the energy distribution as in the case of the Schwarzschild black hole solution.