We study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian (-Delta )^s in bounded open Lipschitz sets in the small order limit s rightarrow 0^+. While it is easy to see that all eigenvalues converge to 1 as s rightarrow 0^+, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with Fourier symbol 2log |xi |. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that L^2-normalized Dirichlet eigenfunctions of (-Delta )^s corresponding to the k-th eigenvalue are uniformly bounded and converge to the set of L^2-normalized eigenfunctions of the logarithmic Laplacian. In order to derive these spectral asymptotics, we establish new uniform regularity and boundary decay estimates for Dirichlet eigenfunctions for the fractional Laplacian. As a byproduct, we also obtain corresponding regularity properties of eigenfunctions of the logarithmic Laplacian.