This paper presents an analysis of the inclusive properties of diffractive deep inelastic scattering events produced in $ep$ interactions at HERA. The events are characterised by a rapidity gap between the outgoing proton system and the remaining hadronic system. Inclusive distributions are presented and compared with Monte Carlo models for diffractive processes. The data are consistent with models where the pomeron structure function has a hard and a soft contribution. The diffractive structure function is measured as a function of $\xpom$, the momentum fraction lost by the proton, of $β$, the momentum fraction of the struck quark with respect to $\xpom$, and of $Q^2$. The $\xpom$ dependence is consistent with the form \xpoma where $a~=~1.30~±~0.08~(stat)~^{+~0.08}_{-~0.14}~(sys)$ in all bins of $β$ and $Q^2$. In the measured $Q^2$ range, the diffractive structure function approximately scales with $Q^2$ at fixed $β$. In an Ingelman-Schlein type model, where commonly used pomeron flux factor normalisations are assumed, it is found that the quarks within the pomeron do not saturate the momentum sum rule. ; Comment: 36 pages, latex, 11 figures appended as uuencoded file