We have measured the dissipation associated with superfluid turbulence in a nonuniform channel of circular cross section in order to further explore the discrepancies with theory discovered previously in a nonuniform rectangular channels. Our data show that the two turbulent states supported in uniform circular channels (TI andTII) are also present in our experiments for either diverging or converging flow, indicating that this two-state structure is quite robust. The transition from laminar flow to theTI state and the transition from theTI to theTII state each occur at stationary turbulent fronts in the channel. One consequence of this phenomenon is that theTI/TII transition is substantially broadened. Our data for theTI state, when analysed assuming local uniformity, are in excellent agreement with previous results from uniform channels. The broadenedTI/TII transition severely restricts our experimental access to theTII state. Nevertheless, our data are consistent with the same kind of modification to the vortex line density found for the nonuniform rectangular channels.