A detailed description is given of the analysis of neutral kaons decaying t
o pi (+)pi (-), based on the complete set of data collected with the CPLEAR
experiment. Using a novel approach involving initially strangeness-tagged
K-0 and (K) over bar (0), the time-dependent decay-rate asymmetry has been
measured. This asymmetry, resulting from the interference between the K-S a
nd K-L decay amplitudes, has enabled both the magnitude and phase of the CP
-violation parameter, eta (+-), to be measured, with a precision comparable
to that of the current world-average values.