Since stream power ω and sediment transport rate i are different values of the same physical quantity, namely, the time rate of energy supply and dissipation, it is rational to relate one to the other. The experimental relation has been difficult to interpret because of the spurious curvature of log-log plots in which a constant threshold stream power of zero is involved. The substitution of an excess power ω − ω0 removes this curvature, and existing data on laboratory bed load transport rate measurements ib suggest a general empirical relation: ib ∝ (ω − ω0)[(ω − ω0);0]½. Existing laboratory data have also shown clearly that at any given value of ω − ω0 the bedload transport rate ib decreases as an inverse function of the ratio flow depth to grain size Y/D. The East Fork River (Wyoming) project has recently enabled bed load sampling devices to be calibrated, so reasonably reliable measurements can be made in natural rivers. The uncertainties in the measurement of the corresponding river power are discussed, and a simple data reliability test is suggested. Data covering three seasons collected from both Snake and Clearwater rivers appear to be reliable. Though there is much scatter due to day variations in the river conditions, these data, together with data on an imtermediate scale from East Fork River and on a small laboratory scale, conform with startling consistency to the following general empirical relation: ib/(ω − ω0) ≈ [(ω − ω0)/ω0]½(Y/D)−⅔ over a 2 million-fold range of stream discharge. The degree of consistency of the above empirical relation with the theoretical relation deduced previously (Bagnold, 1973) is discussed, as are also some morphological implications of the dependence of ib on the depth to grain size ratio Y/D.