Conductivity exhibiting power-law frequency response with an exponent of unity leads to frequency-independent dielectric loss. Such constant-loss (CL) behavior is not physically realizable over a nonzero frequency range, and approximate expressions that have been used to represent it are inconsistent with the Kronig-Kramers relations. Response models are proposed and investigated that do satisfy these relations and can lead to very close approximation to CL over many frequency decades, as often observed at low temperatures in ionic conductors such as glasses. Apparent CL response is shown to arise from the series connection of a constant-phase complex-power-law element (CPE), with exponent delta (0 < delta <1), and a frequency-independent dielectric constant, epsilon (U). Two physically disparate situations can lead to such a series connection. The first involves bulk CPE response in series with an electrode-related, double-layer blocking capacitance involving a dielectric constant epsilon (S). Then, apparent CL behavior may be associated with localized ionic motion in the bulk of the material. The second (mirror-image) situation involves CPE response associated with ionic motion in or at an electrode in series with a capacitance such as the bulk high-frequency-limiting total dielectric constant epsilon (infinity) or the pure-dielectric quantity epsilon (D infinity). The present model is used to simultaneously fit both the real and imaginary parts of data derived from measurements on a sodium-trisilicate glass at 122 K. This data set exhibits power-law nearly constant loss for epsilon (')(omega) and apparent CL for epsilon (')(omega). The magnitude of the CL closely satisfies a simple equation involving only delta and epsilon (U). Further, for the electrode-power-law situation, estimated values of limiting-high-frequency dielectric constants turn out to be more consistent with bulk values established at much higher temperatures where nearly constant loss is no longer a dominant part of the response. Data at -0.5 degreesC are also analyzed with a more complicated composite model, one that is a generalization of both of the above approaches, and nearly constant loss bulk, not electrode, power-law effects in both epsilon (')(omega) and epsilon (')(omega) are isolated and quantified. For this data set it is shown that electrode effects are important at both ends of the frequency range.