Quantum fluctuations of the gravitational field in the early Universe, amplified by inflation, produce a primordial gravitational-wave background across a broad frequency band. We derive constraints on the spectrum of this gravitational radiation, and hence on theories of the early Universe, by combining experiments that cover 29 orders of magnitude in frequency. These include Planck observations of cosmic microwave background temperature and polarization power spectra and lensing, together with baryon acoustic oscillations and big bang nucleosynthesis measurements, as well as new pulsar timing array and ground-based interferometer limits. While individual experiments constrain the gravitational-wave energy density in specific frequency bands, the combination of experiments allows us to constrain cosmological parameters, including the inflationary spectral index nt and the tensor-to-scalar ratio r. Results from individual experiments include the most stringent nanohertz limit of the primordial background to date from the Parkes Pulsar Timing Array, ΩGW(f) < 2.3 × 10−10. Observations of the cosmic microwave background alone limit the gravitational-wave spectral index at 95% confidence to nt ≲ 5 for a tensor-to-scalar ratio of r = 0.11. However, the combination of all the above experiments limits nt < 0.36. Future Advanced LIGO observations are expected to further constrain nt < 0.34 by 2020. When cosmic microwave background experiments detect a nonzero r, our results will imply even more stringent constraints on nt and, hence, theories of the early Universe.