In the interior of stars, convective turbulence (1) is characterized by eddies of all sizes and (2) is compressible (the sound speed is finite). The mixing-length theory (MLT) approximates the first by assuming that there is only one large eddy, and the second by treating turbulence as incompressible. Here, we correct the first MLT approximation. Using two modern theories of turbulence, we compute the full spectrum of eddies and propose a new formula for the turbulent convective flux F(c) to replace the corresponding MLT expression. At high convective efficiencies, the new F(c) is up to 10 times larger than F(c)(MLT), a result recently confirmed by direct numerical simulation of turbulent convection. We also calculate the turbulent pressure exactly. As for the second MLT approximation, we retain the incompressibility condition but improve over the MLT by employing two expressions for the mixing length: (1) the standard LAMBDA = alpha-H(p) and (2) a new expression free of adjustable parameters, LAMBDA = z, where z is the distance to the top of the convective zone. We apply the new model to study the evolution of a 1 M . star and of a globular cluster star of 0.8 M .. With LAMBDA = alpha-H(p), the fit to the data requires alpha = 0.7, thus eliminating the inconsistency of an alpha > 1 usually required by the MLT formalism. With LAMBDA = z, we fit the effective temperature of the Sun within 0.2% without free parameters. Implications for helioseismology are briefly discussed. We also discuss the evolutionary tracks of a solar red giant, a globular cluster main sequence, and a red giant of 0.8 M .. The new model predicts a subatmospheric density inversion larger than that given by the MLT. Also, the observational turnoff of globular clusters requires a larger mass and thus a younger age, a result of possible cosmological interest. Future applications of the new model are discussed, together with suggestions of how to incorporate the new model in stellar evolutionary codes. Although much work remains to be done to assess the compatibility of the new model with a wider set of observational data, as well as to treat compressibility effects, we believe that the three new ingredients, (1) the full spectrum of turbulent eddies, (2) the absence of free parameters in the LAMBDA = z model, and (3) the fit to the Sun's data, have resulted in a model considerably more robust and complete than the MLT.
