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Of central importance to meta-analysis is the estimation of unknown meta-analytic quantities, such as the overall mean or between-studies standard deviation. Though not always the case, recent works in Bayesian meta-analysis have shifted focus from non-informative to a more informative class of prior distributions for between-studies heterogeneity. We continue this trend and propose a new weakly informative prior distribution for a between-studies standard deviation parameter in Bayesian meta-analysis. In contrast to select existing weakly informative or informative prior distributions, the data-driven quality of this prior distribution uses only data that pertain to the meta-analysis at hand. Specifically, the new prior distribution uses a folded noncentral t distribution with a frequentist estimate of the between-studies standard deviation as the noncentrality parameter.