Given alpha is an element of [0, 2) and f is an element of L(2) ((0, T) x (0, 1)), we derive new Carleman estimates for the degenerate parabolic problem w(t) + (x(alpha)w(x))(x) = f, where (t, x). (0, T) x (0, 1), associated to the boundary conditions w(t, 1) = 0 and w(t, 0) = 0 if 0 <= alpha < 1 or (x(alpha)w(x))(t, 0) = 0 if 1 <= alpha < 2. The proof is based on the choice of suitable weighted functions and Hardy- type inequalities. As a consequence, for all 0 <= alpha < 2 and omega subset of subset of (0, 1), we deduce null controllability results for the degenerate one-dimensional heat equation u(t) - (x(alpha)u(x))(x) = h chi(omega) with the same boundary conditions as above.