If the dark matter is produced in the early universe prior to Big Bang nucleosynthesis, a modified cosmological history can drastically affect the abundance of relic dark matter particles. Here, we assume that an additional species to radiation dominates at early times, causing the expansion rate at a given temperature to be larger than in the standard radiation-dominated case. We demonstrate that, if this is the case, dark matter production via freeze-in (a scenario when dark matter interacts very weakly, and is dumped in the early universe out of equilibrium by decay or scattering processes involving particles in the thermal bath) is dramatically suppressed. We illustrate and quantitatively and analytically study this phenomenon for three different paradigmatic classes of freeze-in scenarios. For the frozen-in dark matter abundance to be as large as observations, couplings between the dark matter and visible-sector particles must be enhanced by several orders of magnitude. This sheds some optimistic prospects for the otherwise dire experimental and observational outlook of detecting dark matter produced by freeze-in.