  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 $\beta$, 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
$a~=~1.30~\pm~0.08~(stat)~^{+~0.08}_{-~0.14}~(sys)$ in all bins of $\beta$ and
$Q^2$. In the measured $Q^2$ range, the diffractive structure function
approximately scales with $Q^2$ at fixed $\beta$. 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.
