We report the discovery of PSR J0952-0607, a 707 Hz binary millisecond pulsar that is now the fastest-spinning neutron star known in the Galactic field (i. e., outside of a globular cluster). PSR J0952-0607 was found using LOFAR at a central observing frequency of 135 MHz, well below the 300 MHz to 3 GHz frequencies typically used in pulsar searches. The discovery is part of an ongoing LOFAR survey targeting unassociated Fermi-Large Area Telescope.-ray sources. PSR J0952-0607 is in a 6.42 hr orbit around a very low-mass companion (M greater than or similar to 0.02 M), and we identify a strongly variable optical source, modulated at the orbital period of the pulsar, as the binary companion. The light curve of the companion varies by 1.6 mag from r = 22.2 at maximum to r > 23.8, indicating that it is irradiated by the pulsar wind. Swift observations place a 3s upper limit on the 0.3 -10 keV X-ray luminosity of L-X < 1.1 x 10(31) erg s(-1) (using the 0.97 kpc distance inferred from the dispersion measure). Though no eclipses of the radio pulsar are observed, the properties of the system classify it as a black widow binary. The radio pulsed spectrum of PSR J0952-0607, as determined through flux density measurements at 150 and 350 MHz, is extremely steep with alpha similar to-3 (where S mu n(alpha)). We discuss the growing evidence that the fastest-spinning radio pulsars have exceptionally steep radio spectra, as well as the prospects for finding more sources like PSR J0952-0607.