We present the radial-velocity follow-up of two Kepler planetary transiting candidates (KOI-189 and KOI-686) carried out with the SOPHIE spectrograph at the Observatoire de Haute Provence. These data promptly discard these objects as viable planet candidates and show that the transiting objects are in the regime of very low-mass stars, where a strong discrepancy between observations and models persists for the mass and radius parameters. By combining the SOPHIE spectra with the Kepler light curve and photometric measurements found in the literature, we obtain a full characterization of the transiting companions, their orbits, and their host stars. The two companions are in significantly eccentric orbits with relatively long periods (30 days and 52.5 days), which makes them suitable objects for a comparison with theoretical models, since the effects invoked to understand the discrepancy with observations are weaker for these orbital distances. KOI-189 B has a mass M = 0.0745 +/- 0.0033 Msun and a radius R = 0.1025 +/- 0.0024 Rsun. The density of KOI-189 B is significantly lower than expected from theoretical models for a system of its age. We explore possible explanations for this difference. KOI-189 B is the smallest hydrogen-burning star with such a precise determination of its fundamental parameters. KOI-686 B is larger and more massive (M = 0.0915 +/- 0.0043 Msun; R = 0.1201 +/- 0.0033 Rsun), and its position in the mass-radius diagram agrees well with theoretical expectations.
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