The production of a primordial stochastic gravitational-wave background by processes occuring in the early Universe is expected in a broad range of models. Observing this background would open a unique window onto the Universes evolutionary history. Probes like the Cosmic Microwave Background (CMB) or the Baryon Acoustic Oscillations (BAO) can be used to set upper limits on the stochastic gravitational-wave background energy density $Omega_{GW}$ for frequencies above $10^{-15}$ Hz. We perform a profile likelihood analysis of the Planck CMB temperature anisotropies and gravitational lensing data combined with WMAP low-$ell$ polarization, BAO, South Pole Telescope and Atacama Cosmology Telescope data. We find that $Omega_{GW}h_{0}^{2} < 3.8 times 10^{-6}$ at 95% confidence level for adiabatic initial conditions which improves over the previous limit by a factor 2.3. Assuming that the primordial gravitational waves have been produced by a network of cosmic strings, we have derived exclusion limits in the cosmic string parameter space. If the size of the loops is determined by gravitational back-reaction, string tension values greater than $sim 4 times 10^{-9}$ are excluded for a reconnection probability of $10^{-3}$.
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