  We present a parametric strong lensing model of the cluster Abell 2218 based
on HST ACS data. We constrain the lens model using 8 previously known multiply
imaged systems, of which 7 have spectroscopically confirmed redshifts. In
addition, we propose five candidate multiply imaged systems and estimate their
redshifts using our mass model. The model parameters are optimized in the
source plane by a bayesian Monte Carlo Markov Chain as implemented in the the
publicly available software Lenstool. We find rms=0."12 for the scatter of the
sources in the source plane, which translates into rms=1."49 between the
predicted and measured image positions in the image plane. We find that the
projected mass distribution of Abell 2218 is bimodal, which is supported by an
analysis of the light distribution. We also find evidence for two structures in
velocity space, separated by ~1000 km/s, corresponding to the two large scale
dark matter clumps. We find that the lensing constraints can not be well
reproduced using only dark matter halos associated with the cluster galaxies,
but that the dark matter is required to be smoothly distributed in large scale
halos. At 100" the enclosed projected mass is 3.8e14 solar masses. At that
radius, the large scale halos contribute ~85% of the mass. We find that the
model is not very sensitive to the fainter galaxy sized halos, unless they
locally perturb a given multiply imaged system. Therefore, dark galaxy sized
substructure can be reliably constrained only if it locally perturbs one of the
systems. In an appendix we give a self-contained description of the parametric
profile we use, the dual pseudo isothermal elliptical mass distribution (dPIE).
This profile is a two component pseudo isothermal mass distribution (PIEMD)
with both a core radii and a scale radii. (Abridged)
