The structure of shell helium-burning stars with a carbon-oxygen core of relatively small mass is studied under the assumption of thermal equilibrium, and a general picture of model properties is presented as a function of the core mass and total model mass. The envelope mass DELTA-M necessary for stable burning is a decreasing function of the core mass M(CO): DELTA-M = 0.30 M. for helium main-sequence models (M(CO) = 0); DELTA-M = 0.10 M. for M(CO) = 0.3 M. and, thereafter, it decreases rapidly to DELTA-M = 0.010 M. at M(CO) = 0.6 M.. The helium-burning equivalent of the Schonberg-Chandrasekhar limit (SCHe) is effective only for M(CO) greater-than-or-equal-to 0.43 M. and for a stellar mass M greater-than-or-equal-to 1.5 M.. The corresponding upper limit on the mass fraction of a nondegenerate, isothermal core is a gradually decreasing function of the stellar mass (or, equivalently, of the core mass), dropping from 29% at M(CO) = 0.45 M. to 25% at M(CO) = 0.6 M.. We find also a lower limit on the mass fraction in an electron-degenerate core for M(CO) < 0.6 M.. When the mass fraction in the core decreases below this limit, the core "evaporates" through heating by shell burning; this phenomenon is the exact opposite of the gravothermal catastrophe triggered when the SCHe limit on a nondegenerate core is exceeded. The minimum critical mass fraction is also a decreasing function of the core mass, changing from 23% at M(CO) = 0.45 M. to 14% at M(CO) = 0.5 M.. The evaporation limit is smaller than the SCHe limit, and between these two limits, the core configurations are bimodal. For M(CO) less-than-or-equal-to 0.43 M. and M less-than-or-equal-to 1.5 M., neither of the two limits exists and the transition between the nondegenerate, isothermal configuration and the degenerate configuration is smooth and continuous. The envelope structure also varies with the core configuration. A model star with a nondegenerate core has a compact and hot envelope, and stays very close to the helium main sequence in the H-R diagram. Models with a degenerate core change their appearance depending upon the core and envelope mass. More massive cores can support more extended and cooler envelopes. A core of mass larger than M(CO) congruent-to 0.5 M. is required to realize a red giant envelope with deep convection. Models with M greater-than-or-equal-to 0.7 M. become red giants when the core mass exceeds 0.5 M.. However, there exists an upper bound on the envelope mass for red giants. This limit increases with the core mass, being DELTA-M = 1.07 M. for M(CO) = 0.5 M. and 3.03 M. for M(CO) = 0.6 M.. When the envelope is more massive than this limit, a model contracts to become a blue giant. In contrast, models with M(CO) less-than-or-equal-to 0.3 M. remain in the vicinity of the helium main sequence, regardless of the stellar mass. Consequently, a low-mass helium star traverses the H-R diagram toward the red while the core grows from M(CO) congruent-to 0.4 M. to 0.5 M.. For models of mass between M congruent-to 0.7 M. and 1.5 M., these tracks lie in a narrow range of luminosity around 3 x 10(3) L., and they spend several times 10(5) yr in this range. The present results may be relevant to known hydrogen-deficient stars, formed through mass loss or merging in close binary systems. Finally, based on these results, we show that the red giant structure (a very large envelope and a compact core separated by a burning shell) is attributable to the power-law distribution of density which is forced when envelope matter is of sufficiently high entropy that self-gravity can be ignored relative to the gravity produced by the inner core over a distance which is many times larger than the core radius. In the course of evolution, the conditions necessary for such a configuration are realized through (1) the occurrence of a gravothermal catastrophe in the core, and by (2) the presence of a buffer zone within which there is a sufficient entropy gradient to prevent the envelope from following the catastrophe.
