Cepheid stars play a considerable role as extragalactic distances indicators, thanks to the simple empirical relation between their pulsation period and their luminosity. They overlap with that of secondary distance indicators, such as Type Ia supernovae, whose distance scale is tied to Cepheid luminosities. However, the period-luminosity (P-L) relation still lacks a calibration to better than 5%. Using an original combination of interferometric astrometry with optical and ultraviolet spectroscopy, we measured the geometrical distance d = 720.35 +/- 7.84 pc of the 3.33 day period Cepheid V1334 Cyg with an unprecedented accuracy of +/- 1%, providing the most accurate distance for a Cepheid. Placing this star in the P-L diagram provides an independent test of existing P-L relations. We show that the secondary star has a significant impact on the integrated magnitude, particularly at visible wavelengths. Binarity in future high-precision calibrations of the P-L relations is not negligible, at least in the short-period regime. Subtracting the companion flux leaves V1334 Cyg in marginal agreement with existing photometric-based P-L relations, indicating either an overall calibration bias or a significant intrinsic dispersion at a few percent level. Our work also enabled us to determine the dynamical masses of both components, M-1 = 4.288 +/- 0.133 M-circle dot (Cepheid) and M-2 = 4.040 +/- 0.048 M-circle dot (companion), providing the most accurate masses for a Galactic binary Cepheid system.