We introduce a distance-based phylogeny reconstruction method called "weighted neighbor joining," or "Weighbor" for short. As in neighbor joining, two taxa are joined in each iteration; however, the Weighbor criterion for choosing a pair of taxa to join takes into account that errors in distance estimates are exponentially larger for longer distances. The criterion embodies a likelihood function on the distances, which are modeled as correlated Gaussian random variables with different means and variances, computed under a probabilistic model for sequence evolution. The Weighbor criterion consists of two terms, an additivity term and a positivity term, that quantify the implications of joining the pair. The first term evaluates deviations from additivity of the implied external branches, while the second term evaluates confidence that the implied internal branch has a positive branch length. Compared with maximum-likelihood phylogeny reconstruction, Weighbor is much faster, while building trees that are qualitatively and quantitatively similar. Weighbor appears to be relatively immune to the "long branches attract" and "long branch distracts" drawbacks observed with neighbor joining, BIONJ, and parsimony.