By developing an analogy with the measure of association in a fourfold contingency table, a method is given of constructing a one-parameter class of bivariate distributions from given margins. This class contains the known boundary distributions and the member corresponding to independent random variables. The class resulting from standard normal margins is compared with the standard normal bivariate distribution, and it is found that the two joint distributions and the two conditional distributions each agree tolerably well. Estimation of the parameter is illustrated by an example showing a computational advantage of this class of distributions.