We calculate the dispersion measure (DM) contributed by the intergalactic medium (IGM) to the total measured DM for fast radio bursts (FRBs). We use the MareNostrum Instituto de Ciencias del Espacio (MICE) Onion Universe simulation to track the evolution of the dark matter particle density over a large range of redshifts. We convert this dark matter particle number density to the corresponding free electron density and then integrate it to find the DM as a function of redshift. This approach yields an intergalactic DM of pc cm−3, with the large errors representative of the structure in the IGM. We place limits on the redshifts of the current population of observed FRBs. We also use our results to estimate the host galaxy contribution to the DM for the first repeater, FRB 121102, and show that the most probable host galaxy DM contribution, pc cm−3, is consistent with the estimate made using the Balmer emission lines in the spectrum of the host galaxy, pc cm−3. We also compare our predictions for the host galaxy contribution to the DM for the observations of FRB 180924 and FRB 190523, both of which have been localized to a host galaxy.