  The dimer model on a strip is considered as a Yang-Baxter \mbox{integrable}
six vertex model at the free-fermion point with crossing parameter
$\lambda=\tfrac{\pi}{2}$ and quantum group invariant boundary conditions. A
one-to-many mapping of vertex onto dimer configurations allows for the solution
of the free-fermion model to be applied to the anisotropic dimer model on a
square lattice where the dimers are rotated by $45\degree$ compared to their
usual orientation. In a suitable gauge, the dimer model is described by the
Temperley-Lieb algebra with loop fugacity $\beta=2\cos\lambda=0$. It follows
that the model is exactly solvable in geometries of arbitrary finite size. We
establish and solve transfer matrix inversion identities on the strip with
arbitrary finite width $N$. In the continuum scaling limit, in sectors with
magnetization $S_z$, we obtain the conformal weights
$\Delta_{s}=\big((2-s)^2-1\big)/8$ where $s=|S_z|+1=1,2,3,\ldots$. We further
show that the corresponding finitized characters $\chit_s^{(N)}(q)$ decompose
into sums of $q$-Narayana numbers or, equivalently, skew $q$-binomials. In the
particle representation, the local face tile operators give a representation of
the fermion algebra and the fermion particle trajectories play the role of
nonlocal degrees of freedom. We argue that, in the continuum scaling limit,
there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation
operator $L_0$. This confirms that, with quantum group invariant boundary
conditions, the dimer model gives rise to a {\em logarithmic} conformal field
theory with central charge $c=-2$, minimal conformal weight
$\Delta_{\text{min}}=-\frac{1}{8}$ and effective central charge
$c_{\text{eff}}=1$.Our analysis of the structure of the ensuing rank 2 modules
indicates that the familiar staggered $c=-2$ modules appear as submodules.
