  This is the first paper in a series that present a multi-wavelength analysis
of the archetype Ultra-Luminous InfraRed Galaxy (ULIRG) IRAS FSC10214+4724, a
gravitationally lensed, starburst/AGN at z=2.3. Here we present a new lens
model and spatially-resolved radio data, as well as a deep HST F160W map. The
lens modelling employs a Bayesian Markov Chain Monte Carlo algorithm with
extended-source, forward ray-tracing. Using these high resolution HST, MERLIN
and VLA maps, the algorithm allows us to constrain the level of distortion to
the continuum spectral energy distribution resulting from emission components
with differing magnification factors, due to their size and proximity to the
caustic. Our lens model finds the narrow line region (NLR), and by proxy the
active nucleus, is preferentially magnified. This supports previous claims that
preferential magnification could mask the expected polycyclic aromatic
hydrocarbon spectral features in the Spitzer mid-infrared spectrum which
roughly trace the star-forming regions. Furthermore, we show the
arc-to-counter-image flux ratio is not a good estimate of the magnification in
this system, despite its common use in the IRAS FSC10214+4724 literature. Our
lens modelling suggests magnifications of \mu ~ 15-20+-2 for the HST F814W,
MERLIN 1.7 GHz and VLA 8 GHz maps, significantly lower than the canonical
values of \mu = 50-100 often used for this system. Systematic errors such as
the dark matter density slope and co-location of stellar and dark matter
centroids dominate the uncertainties in the lens model at the 40 percent level.
