  Direct detection of low-frequency gravitational waves ($10^{-9} - 10^{-8}$
Hz) is the main goal of pulsar timing array (PTA) projects. One of the main
targets for the PTAs is to measure the stochastic background of gravitational
waves (GWB) whose characteristic strain is expected to approximately follow a
power-law of the form $h_c(f)=A (f/\hbox{yr}^{-1})^{\alpha}$, where $f$ is the
gravitational-wave frequency. In this paper we use the current data from the
European PTA to determine an upper limit on the GWB amplitude $A$ as a function
of the unknown spectral slope $\alpha$ with a Bayesian algorithm, by modelling
the GWB as a random Gaussian process. For the case $\alpha=-2/3$, which is
expected if the GWB is produced by supermassive black-hole binaries, we obtain
a 95% confidence upper limit on $A$ of $6\times 10^{-15}$, which is 1.8 times
lower than the 95% confidence GWB limit obtained by the Parkes PTA in 2006. Our
approach to the data analysis incorporates the multi-telescope nature of the
European PTA and thus can serve as a useful template for future
intercontinental PTA collaborations.
