  Context. The pursuit of more realistic spectroscopic modelling and consistent
abundances has led us to begin a new series of papers designed to improve
current solar and stellar abundances of various atomic species. To achieve
this, we have began updating the three-dimensional (3D) non-local thermodynamic
equilibrium (non-LTE) radiative transfer code, Multi3D, and the equivalent
one-dimensional (1D) non-LTE radiative transfer code, MULTI. Aims. We examine
our improvements to these codes by redetermining the solar barium abundance.
Barium was chosen for this test as it is an important diagnostic element of the
s-process in the context of galactic chemical evolution. New Ba II + H
collisional data for excitation and charge exchange reactions computed from
first principles had recently become available and were included in the model
atom. The atom also includes the effects of isotopic line shifts and hyperfine
splitting. Method. A grid of 1D LTE barium lines were constructed with MULTI
and fit to the four Ba II lines available to us in the optical region of the
solar spectrum. Abundance corrections were then determined in 1D non-LTE, 3D
LTE, and 3D non-LTE. A new 3D non-LTE solar barium abundance was computed from
these corrections. Results. We present for the first time the full 3D non-LTE
barium abundance of $A({\rm Ba})=2.27\pm0.02\pm0.01$, which was derived from
four individual fully consistent barium lines. Errors here represent the
systematic and random errors, respectively.
