Context. Estimating the physical states of the surfaces of fast-rotating stars is challenging due to several intrinsic processes, which include radiative flux inhomogeneities on the photosphere induced by rotation and circumstellar signatures in their spectra. The analysis of their spectra ultimately requires the use of synthetic grids of spectra accounting for all these physical processes. Aims. In this paper, we present the 'von ZeiPEl's code for gravity darKening specTRal synthesis'(ZPEKTR) code, which is designed to perform the spectral synthesis of fast-rotating stars, accounting for gravity darkening, limb-darkening effects in the continuum and geometrical deformation induced by fast rotation. Methods. We consider colatitudinal temperature and surface-gravity variations, assuming both the classical prescription developed by von Zeipel and the new formulation by Espinosa-Lara. The code runs either with a rectangular or a triangular mesh on the stellar surface. We compare the temperature and gravitational distribution as a function of the stellar latitude arising from both models. Results. The line profiles of He I 4388, 4471, 4922, and 6678 Å produced with both formalisms are compared at three different rotation rates and illustrate differences in shape and central intensity. We also illustrate the fittings of 31 line spectra of classical Be stars averaged from the Be Stars Observation Survey (BeSOS) database and make a comparison among their apparent physical parameters and ages determined from plane-parallel non-local thermodynamical equilibrium (non-LTE) models and parameters determined from classical von Zeipel models, finding a displacement of more evolved objects towards the zero-age main sequence. We also compare the distributions of projected rotation velocities of these objects obtained with and without the inclusion of gravity-darkening effects with ZPEKTR. Conclusions. We observe a shift of the histogram of rotation velocities calculated accounting for effects of gravity darkening concerning rotation velocities obtained through the fittings with classical plane-parallel non-LTE models. We show that models that do not account for gravity darkening can underestimate the rotation velocity, because the stellar latitudes that contribute the higher velocities are those in the equator with the least radiative flux. We envisage near-future improvements to the code, such as the inclusion of differential rotation and treatment of tidal forces in binary stellar systems.