In this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces W^s_p( {R}), where {p (1,2]} and {s (1+1/p,2)}. This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in W^{{s}-2}_p( {R}), where {{s} (1+1/p,s)}. Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.