Spin-orbit coupling has been conjectured to play a key role in the low-energy electronic structure of Sr2RuO4. By using circularly polarized light combined with spin-and angle-resolved photoemission spectroscopy, we directly measure the value of the effective spin-orbit coupling to be 130 +/- 30 meV. This is even larger than theoretically predicted and comparable to the energy splitting of the d(xy) and d(xz,yz) orbitals around the Fermi surface, resulting in a strongly momentum-dependent entanglement of spin and orbital character in the electronic wavefunction. As demonstrated by the spin expectation value <(S-k) over arrow.(S-k) over arrow > calculated for a pair of electrons with zero total momentum, the classification of the Cooper pairs in terms of pure singlets or triplets fundamentally breaks down, necessitating a description of the unconventional superconducting state of Sr2RuO4 in terms of these newly found spin-orbital entangled eigenstates.