At a density near a few ×107 g cm–3, the subsonic burning in a Type Ia supernova (SN) enters the distributed regime (high Karlovitz number). In this regime, turbulence disrupts the internal structure of the flame, and so the idea of laminar burning propagated by conduction is no longer valid. The nature of the burning in this distributed regime depends on the turbulent Damköhler number (DaT), which steadily declines from much greater than one to less than one as the density decreases to a few ×106 g cm–3. Classical scaling arguments predict that the turbulent flame speed sT, normalized by the turbulent intensity , follows for DaT 1. The flame in this regime is a single turbulently broadened structure that moves at a steady speed, and has a width larger than the integral scale of the turbulence. The scaling is predicted to break down at DaT 1, and the flame burns as a turbulently broadened effective unity Lewis number flame. This flame burns locally with speed sλ and width lλ, and we refer to this kind of flame as a λ-flame. The burning becomes a collection of λ-flames spread over a region approximately the size of the integral scale. While the total burning rate continues to have a well-defined average, , the burning is unsteady. We present a theoretical framework, supported by both one-dimensional and three-dimensional numerical simulations, for the burning in these two regimes. Our results indicate that the average value of sT can actually be roughly twice for DaT 1, and that localized excursions to as much as 5 times can occur. We also explore the properties of the individual flames, which could be sites for a transition to detonation when DaT ~ 1. The λ-flame speed and width can be predicted based on the turbulence in the star (specifically the energy dissipation rate ε*) and the turbulent nuclear burning timescale of the fuel τTnuc. We propose a practical method for measuring sλ and lλ based on the scaling relations and small-scale computationally inexpensive simulations. This suggests that a simple turbulent flame model can be easily constructed suitable for large-scale distributed SNe flames. These results will be useful both for characterizing the deflagration speed in larger full-star simulations, where the flame cannot be resolved, and for predicting when detonation occurs.