Decomposing the shear signal into E and B modes properly, i.e. without leakage of B modes into the E-mode signal and vice versa, has been a long-standing problem in weak gravitational lensing. At the two-point level this problem was resolved by developing the so-called ring statistic, and later the COSEBIs; however, extending these concepts to the three-point level is far from trivial. Currently used methods to decompose shear three-point correlation functions (3PCFs) into E and B modes require knowledge of the 3PCF down to arbitrary small scales. This implies that the 3PCF needs to be modelled on scales smaller than the minimum separation of two galaxies and will subsequently be biased towards the model, or, in the absence of a model, the statistic is affected by E-/B-mode leakage (or mixing). In this paper, we derive a new third-order E-/B-mode statistic that performs the decomposition using the 3PCF only on a finite interval, and thereby is free of any E-/B-mode leakage while at the same time relying solely on information from the data. In addition, we relate this third-order ring statistic to the convergence field, thereby enabling a fast and convenient calculation of this statistic from numerical simulations. We note that our new statistic should be applicable to corresponding E-/B-mode separation problems in the cosmic microwave background polarization field.