RNA sequencing allows for simultaneous transcript discovery and quantification, but reconstructing complete transcripts from such data remains difficult. Here, we introduce Bayesembler, a novel probabilistic method for transcriptome assembly built on a Bayesian model of the RNA sequencing process. Under this model, samples from the posterior distribution over transcripts and their abundance values are obtained using Gibbs sampling. By using the frequency at which transcripts are observed during sampling to select the final assembly, we demonstrate marked improvements in sensitivity and precision over state-of-the-art assemblers on both simulated and real data. Bayesembler is available at https://github.com/bioinformatics-centre/bayesembler.