  The spectrometer aboard INTEGRAL, SPI, has the capability to detect the
signature of polarised emission from a bright gamma-ray source. GRB 041219a is
the most intense burst localised by INTEGRAL and is an ideal candidate for such
a study. Polarisation can be measured using multiple events scattered into
adjacent detectors because the Compton scatter angle depends on the
polarisation of the incoming photon. A search for linear polarisation in the
most intense pulse of duration 66 seconds and in the brightest 12 seconds of
GRB 041219a was performed in the 100-350keV, 100-500keV and 100keV-1MeV energy
ranges. The multiple event data from the spectrometer was analysed and compared
with the predicted instrument response obtained from Monte-Carlo simulations
using the GEANT 4 INTEGRAL mass model. The chi^2 distribution between the real
and simulated data as a function of the percentage polarisation and
polarisation angle was calculated for all three energy ranges. The degree of
linear polarisation in the brightest pulse of duration 66s was found to be
63+/-31% at an angle of 70+/-14 degrees in the 100-350keV energy range. The
degree of polarisation was also constrained in the brightest 12s of the GRB and
a polarisation fraction of 96+/-40% at an angle of 60+/-14 degrees was
determined over the same energy range. However, despite extensive analysis and
simulations, a systematic effect that could mimic the weak polarisation signal
could not be definitively excluded. Our results over several energy ranges and
time intervals are consistent with a polarisation signal of about 60% at a low
level of significance (2 sigma). We conclude that the procedure described here
demonstrates the effectiveness of using SPI as a polarimeter, and is a viable
method of measuring polarisation levels in intense gamma--ray bursts.
