The measurement of the velocity field of the plasma at the solar surface is a standard diagnostic tool in observational solar physics. Detailed information about the energy transport as well as on the stratification of temperature, pressure and magnetic fields in the solar atmosphere are encoded in Doppler shifts and in the precise shape of the spectral lines. The available instruments deliver data of excellent quality and precision. However, absolute wavelength calibration in solar spectroscopy was so far mostly limited to indirect methods and in general suffers from large systematic uncertainties of the order of 100 m/s. During the course of this thesis, a novel wavelength calibration system based on a laser frequency comb was deployed to the solar Vacuum Tower Telescope (VTT), Tenerife, with the goal of enabling highly accurate solar wavelength measurements at the level of 1 m/s on an absolute scale. The frequency comb was developed in a collaboration between the Kiepenheuer-Institute for Solar Physics, Freiburg, Germany and the Max Planck Institute for Quantum Optics, Garching, Germany. The efforts cumulated in the new prototype instrument LARS (Lars is an Absolute Reference Spectrograph) for solar precision spectroscopy which is in preliminary scientific operation since~2013. The instrument is based on the high-resolution echelle spectrograph of the VTT for which feed optics based on single-mode optical fibres were developed for this project. The setup routinely achieves an absolute calibration accuracy of 60 cm/s and a repeatability of 2.5 cm/s. An unprecedented repeatability of only 0.32 cm/s could be demonstrated with a differential calibration scheme. In combination with the high spectral resolving power of the spectrograph of 7x10^5 and virtually absent internal scattered light, LARS provides a spectral purity and fidelity that previously was the domain of Fourier-transform spectrometers only. The instrument therefore provides unique capabilities for precision spectroscopy of the Sun and laboratory light sources. The first scientific observations aimed at measuring the accurate wavelengths of selected solar Fraunhofer lines to characterise the so-called convective blue shift and its centre to limb variation. The convective blueshifts were derived with respect to laboratory wavelengths that were obtained from spectral lamps measured with the same instrument. The measurements agree with previous studies but provide a way higher accuracy. The data is only partially compatible with numerical simulations that were published recently. Further measurements were carried out to provide the absolute wavelengths of telluric O2 lines that are commonly used for wavelength calibration. With an accuracy of 1 m/s, these new measurements are two orders of magnitude better than existing data.