We present the results of observations of radial velocity variability in stars with exoplanets. Observations of a number of stars with characteristic stellar magnitudes from 8.54(m) to 10.5(m) were carried out using a new high-resolution (R = 35 000-120 000) fiber-fed spectrograph mounted at the foundation of the 6-m BTA telescope, in the mode of resolutions from R = 45 000 to R = 65 000 with a simultaneous registration of the calibration spectrum of a thorium-argon hollow cathode lamp. The achieved characteristic measurement accuracies ranged from a few to several tens of m s(-1). In the future, the thorium-argon lamp will be supplemented with a Fabri-Perot etalon to reach the radial velocity measurement accuracy in stars up to 1 m s(-1).