The object of this paper is to extend a result of Agnew in [1] for single series to double series, and so enable Tauberian constants between matrix transformations of double sequences to be calculated using the technique that many authors have used previously for single sequences (see, for instance, [2, 3, 8]). This method (see below) gives the best possible Tauberian constants in the form of certain upper limits, and we attempt to calculate these for Cesàro summability with various known Tauberian conditions. In Section 2, we introduce our notation and review earlier work. In Section 3, we prove our main result, and we discuss applications to Cesàro summability in Section 4.