A new full three-body method is introduced to compute the rate of the triple-alpha capture reaction which is the primary source of $^{12}$C in stars. In this work, we combine the Faddeev hyperspherical harmonics and the R-matrix method to obtain a full solution to the three-body $alpha+alpha+alpha$ continuum. Particular attention is paid to the long range effects caused by the pairwise Coulomb interactions. The new rate agrees with the NACRE rate for temperatures greater than 0.07 GK, but a large enhancement at lower temperature is found ($approx 10^{14}$ at 0.02 GK). Our results are compared to previous calculations where additional approximations were made. We show that the new rate does not significantly change the evolution of stars around one solar mass. In particular, such stars still undergo a red-giant phase consistent with observations, and no significant differences are found in the final white dwarfs.
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