  We explicitly worked out the orbital effects induced on the trajectory of a
test particle by the the weak-field approximation of the Kerr-de Sitter metric.
It results that the node, the pericentre and the mean anomaly undergo secular
precessions proportional to k, which is a measure of the non linearity of the
theory. We used such theoretical predictions and the latest observational
determinations of the non-standard precessions of the perihelia of the inner
planets of the Solar System to put a bound on k getting k <= 10^-29 m^-2. The
node rate of the LAGEOS Earth's satellite yields k <= 10^-26 m^-2. The
periastron precession of the double pulsar PSR J0737-3039A/B allows to obtain k
<= 3 10^-21 m^-2. Interpreting k as a cosmological constant \Lambda, it turns
out that such constraints are weaker than those obtained from the
Schwarzschild-de Sitter metric.
