We report the discovery of the millisecond pulsar PSR J2043+1711 in a search of a Fermi Large Area Telescope (LAT) source with no known associations, with the Nancay Radio Telescope. The new pulsar, confirmed with the Green Bank Telescope, has a spin period of 2.38 ms, is relatively nearby (d approx. < 2 kpc) and is in a 1.48-d orbit around a low-mass companion, probably an He-type white dwarf. Using an ephemeris based on Arecibo, Nancay and Westerbork timing measurements, pulsed gamma-ray emission was detected in the data recorded by the Fermi LAT. The gamma-ray light curve and spectral properties are typical of other gamma-ray millisecond pulsars seen with Fermi. X-ray observations of the pulsar with Suzaku and the Swift X-ray Telescope yielded no detection. At 1.4 GHz, we observe strong flux density variations because of interstellar diffractive scintillation; however, a sharp peak can be observed at this frequency during bright scintillation states. At 327 MHz, the pulsar is detected with a much higher signal-to-noise ratio and its flux density is far more steady. However, at that frequency the Arecibo instrumentation cannot yet fully resolve the pulse profile. Despite that, our pulse time-of-arrival measurements have a post-fit residual rms of 2 micro s. This and the expected stability of this system have made PSR J2043+1711 one of the first new Fermi-selected millisecond pulsars to be added to pulsar gravitational wave timing arrays. It has also allowed a significant measurement of relativistic delays in the times of arrival of the pulses due to the curvature of space-time near the companion, but not yet with enough precision to derive useful masses for the pulsar and the companion. Nevertheless, a mass for the pulsar between 1.7 and 2.0 solar Mass can be derived if a standard millisecond pulsar formation model is assumed. In this paper, we also present a comprehensive summary of pulsar searches in Fermi LAT sources with the Nancay Radio Telescope to date.