Gene expression is a fundamental cellular process by which proteins are eventually synthesized based on the information coded in the genes. This process includes four major steps: transcription of the DNA segment corresponding to a gene to mRNA molecules, the degradation of the mRNA molecules, the translation of mRNA molecules to proteins by the ribosome and the degradation of the proteins. We present an innovative quantitative study of the interaction between the gene translation stage and the mRNA degradation stage using large scale genomic data of S. cerevisiae, which include measurements of mRNA levels, mRNA half-lives, ribosomal densities and protein abundances, for thousands of genes. The reported results support the conjecture that transcripts with higher ribosomal density, which is related to the translation stage, tend to have elevated half-lives, and we suggest a novel quantitative estimation of the strength of this relation. Specifically, we show that on average, an increase of 78% in ribosomal density yields an increase of 25% in mRNA half-life, and that this relation between ribosomal density and mRNA half-life is not function specific. In addition, our analyses demonstrate that ribosomal density along the entire ORF, and not in specific locations, has a significant effect on the transcript half-life. Finally, we show that the reported relation cannot be explained by different expression levels among genes. A plausible explanation for the reported results is that ribosomes tend to protect the mRNA molecules from the exosome complexes degrading them; however, additional non-mutually exclusive possible explanations for the reported relation and experiments for their verifications are discussed in the paper.