Plasmonic cavities can be used to control the atom-photon coupling process at the nanoscale, since they provide an ultrahigh density of optical states in an exceptionally small mode volume. Here we demonstrate strong coupling between molecular excitons and plasmonic resonances (so-called plexcitonic coupling) in a film-coupled nanocube cavity, which can induce profound and significant spectral and spatial modifications to the plasmonic gap modes. Within the spectral span of a single gap mode in the nanocube-film cavity with a 3 nm wide gap, the introduction of narrow-band J-aggregate dye molecules not only enables an anticrossing behavior in the spectral response but also splits the single spatial mode into two distinct modes that are easily identified by their far-field scattering profiles. Simulation results confirm the experimental findings, and the sensitivity of the plexcitonic coupling is explored using digital control of the gap spacing. Our work opens up a new perspective to study the strong coupling process, greatly extending the functionality of nanophotonic systems, with the potential to be applied in cavity quantum electrodynamic systems.