A comparison is made of two versions of the acoustic-mismatch theory of the thermal boundary resistance R which occurs at the interface between a pair of solids. The most recent version predicts R ..-->.. 0 as the two solids become identical, while the earlier version indicates a finite R as the interface essentially vanishes. It is shown that the two versions make different assumptions as to what a thermometer measures and that, if appropriate thermometers are used, both versions give the same result. This conclusion is supported by experimental evidence.