We present a probabilistic full-waveform inversion (FWI) approach that infers a geostatistical model along with the subsurface structure. Probabilistic FWI with Markov chain Monte Carlo (MCMC) allows for uncertainty quantification and removes the requirement of having a starting model in the cone of attraction of the assumed correct global minimum. We demonstrate our approach on a synthetic and a field data set. For the latter, we compare the results with deterministic FWI and a cone penetration test. Our results compare equally well with the cone penetration test as the deterministic results do. This is a positive result as for the deterministic inversion almost seven times more data were used than for the probabilistic inversion. Furthermore, the probabilistic FWI was able to converge to the posterior distribution starting from randomly drawn models. However, our uncertainty estimates are too narrow, because the necessarily short Markov chains implied by a computationally costly forward problem and the global nature of the model proposal scheme prevented a full exploration of the posterior probability density function. Without prior information such as borehole logs, the algorithm is only able to infer relative electric conductivity values, because the unknown amplitude of the wavelet and the mean of the conductivity are strongly correlated. This study clearly demonstrates the feasibility of probabilistic FWI and highlights the advantages and disadvantages of the approach.