  Diffractive scattering of $\gamma^* p \to X + N$, where $N$ is either a
proton or a nucleonic system with $M_N~<~4$~GeV has been measured in deep
inelastic scattering (DIS) at HERA. The cross section was determined by a novel
method as a function of the $\gamma^* p$ c.m. energy $W$ between 60 and 245~GeV
and of the mass $M_X$ of the system $X$ up to 15~GeV at average $Q^2$ values of
14 and 31~GeV$^2$. The diffractive cross section $d\sigma^{diff} /dM_X$ is,
within errors, found to rise linearly with $W$. Parameterizing the $W$
dependence by the form $d\sigma^{diff}/dM_X \propto
(W^2)^{(2\overline{\mbox{$\alpha_{_{I\hspace{-0.2em}P}}$}} -2)}$ the DIS data
yield for the pomeron trajectory
$\overline{\mbox{$\alpha_{_{I\hspace{-0.2em}P}}$}} = 1.23 \pm 0.02(stat) \pm
0.04 (syst)$ averaged over $t$ in the measured kinematic range assuming the
longitudinal photon contribution to be zero. This value for the pomeron
trajectory is substantially larger than
$\overline{\mbox{$\alpha_{_{I\hspace{-0.2em}P}}$}}$ extracted from soft
interactions. The value of $\overline{\mbox{$\alpha_{_{I\hspace{-0.2em}P}}$}}$
measured in this analysis suggests that a substantial part of the diffractive
DIS cross section originates from processes which can be described by
perturbative QCD. From the measured diffractive cross sections the diffractive
structure function of the proton $F^{D(3)}_2(\beta,Q^2,
\mbox{$x_{_{I\hspace{-0.2em}P}}$})$ has been determined, where $\beta$ is the
momentum fraction of the struck quark in the pomeron. The form $F^{D(3)}_2 =
constant \cdot (1/ \mbox{$x_{_{I\hspace{-0.2em}P}}$})^a$ gives a good fit to
the data in all $\beta$ and $Q^2$ intervals with $a = 1.46 \pm 0.04 (stat) \pm
