  The timing properties of the millisecond pulsar PSR J1939+2134 -- very high
rotation frequency, very low time derivative of rotation frequency, no timing
glitches and relatively low timing noise -- are responsible for its exceptional
timing stability over decades. It has been timed by various groups since its
discovery, at diverse radio frequencies, using different hardware and analysis
methods. Most of this timing data is now available in the public domain in two
segments, which have not been combined so far. This work analyzes the combined
data by deriving uniform methods of data selection, derivation of Dispersion
Measure (DM), accounting for correlation due to "red" noise, etc. The timing
noise of this pulsar is very close to a sinusoid, with a period of
approximately $31$ years. The main results of this work are (1) The clock of
PSR J1939+2134 is stable at the level of almost one part in $10^{15}$ over
about $31$ years, (2) the power law index of the spectrum of electron density
fluctuations in the direction of PSR J1939+2134 is $3.86 \pm 0.04$, (3) a Moon
sized planetary companion, in an orbit of semi major axis about $11$
astronomical units and eccentricity $\approx 0.2$, can explain the timing noise
of PSR J1939+2134, (4) Precession under electromagnetic torque with very small
values of oblateness and wobble angle can also be the explanation, but with
reduced confidence, and (5) there is excess timing noise of about $8$ $\mu$s
amplitude during the epochs of steepest DM gradient, of unknown cause.
