A necessary and sufficient condition of regularity of $(0,1,\cdots,m-2,m)$ interpolation on the zeros of Laguerre polynomials $L^{(α)}_n(x) (α≥-1)$ in a manageable form is established. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, it is shown that, if the problem of $(0,1,\cdots,m-2,m)$ interpolation has an infinity of solutions, then the general form of the solutions is $f_0(x)+Cf_1(x)$ with an arbitrary constant $C$.