The dielectric spectrum of the organic glass former 1,1'-bis (p-methoxyphenyl) cyclohexane was measured over a range of temperatures and pressures, corresponding to a variation of the structural relaxation time, tau(alpha), by 8 decades. The temperature dependence of tau(alpha) corresponded to a fragility equal to 72, which is consistent with the correlation of same with the shape (breadth) of the relaxation function. The dependence on pressure of tau(alpha) could be described as a simple activated process, with a pressure-independent activation volume equal to similar to230 cm(3)/mol, varying inversely with temperature. The pressure coefficient of the glass temperature, T-g, was 240 K/GPa. At frequencies beyond the structural relaxation peak, there is a second, thermally activated process, having an activation energy=74.2 kJ/mol. The relaxation times for this process were invariant to pressure. Extrapolation of these relaxation times (measured below the T-g) intersects the ambient pressure structural relaxation data at a temperature, T-beta=268 K. This is about equal to the temperature, T-B, at which the structural relaxation times deviated from a single Vogel-Fulcher-Tamman curve. In this respect, the secondary process exhibits the properties of a Johari-Goldstein relaxation. However, since T-beta varies with pressure, it remains to be seen whether its equivalence to T-B is maintained at elevated pressure.