It is now well established that at a superconductor/ferromagnet (S/F) interface an unconventional superconducting state arises in which the pairing is odd-frequency. The hallmark signature of this superconducting state is generally understood to be an enhancement of the electronic density of states (DoS) at subgap energies close to the S/F interface. However, here we show that an odd frequency state can be present even if the DoS is fully gapped. As an example, we show that this is the case in the pioneering S/FI (where FI is a insulating ferromagnet) tunneling experiments of Meservey and Tedrow, and we derive a generalized analytical criterium to describe the effect of odd-frequency pairing on the DoS. Finally, we propose a simple experiment in which odd-frequency pairing in a Zeeman-split superconductor can be unambiguously detected via the application of an external magnetic field.