We present high-precision relative radial velocities of the very low mass star VB 10 that were obtained over a time span of 0.61 years as part of an ongoing search for planets around stars at the end of the main sequence. The radial velocities were measured from high-resolution near-infrared spectra obtained using the CRIRES instrument on the Very Large Telescope with an ammonia gas cell. The typical internal precision of the measurements is 10 m s^–1. These data do not exhibit significant variability and are essentially constant at a level consistent with the measurement uncertainties. Therefore, we do not detect the radial velocity variations of VB 10 expected due to the presence of an orbiting giant planet similar to that recently proposed by Pravdo & Shaklan based on apparent astrometric perturbations. In addition, we do not confirm the ~1 km s^–1 radial velocity variability of the star tentatively detected by Zapatero Osorio and colleagues with lower precision measurements. Our measurements rule out planets with M _p  > 3 M _Jup and the orbital period and inclination suggested by Pravdo & Shaklan at better than 5σ confidence. We conclude that the planet detection claimed by Pravdo & Shaklan is spurious on the basis of this result. Although the outcome of this work is a non-detection, it illustrates the potential of using ammonia cell radial velocities to detect planets around very low mass stars.