  We present the Final Analysis of the European Large Area ISO Survey (ELAIS)
15 $\mu$m observations, carried out with the ISOCAM instrument on board the
Infrared Space Observatory (ISO).
  The data reduction method, known as LARI method, is based on a mathematical
model of the detector's behaviour and was specifically designed for the
detection of faint sources in ISO-CAM/PHOT data. The method is fully
interactive and leads to very reliable and complete source lists.
  The resulting catalogue includes 1923 sources detected with $S/N > 5$ in the
{0.5 -- 100 mJy} flux range and over an area of 10.85 {deg$^2$} split into four
fields, making it the largest non-serendipitous extragalactic source catalogue
obtained to date from ISO data.
  This paper presents the concepts underlying the data reduction method
together with its latest enhancements. The data reduction process, the
production and basic properties of the resulting catalogue are discussed. The
catalogue quality is assessed by means of detailed simulations, optical
identifications and comparison with previous analyses.
