An analogue of the Riemannian Geometry for an ultrametric Cantor set .C;d/ is described using the tools of Noncommutative Geometry. Associated with.C;d/ is a weighted rooted tree, its Michon tree (29). This tree allows to define a family of spectral triples .CLip.C/; H;D/using the ` 2 -space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here CLip.C/ denotes the space of Lipschitz contin- uous functions on .C;d/. The family of spectral triples is indexed by the space of choice functions, which is shown to be the analogue of the sphere bundle of a Riemannian manifold. The Connes metric coming from the family of these spectral triples allows to recover the metric onC . The corresponding� -function is shown to have abscissa of convergence,s0, equal to the upper box dimension of.C;d/. Taking the residue at this singularity leads to the definition of a canonical probability measure onC , which in certain cases coincides with the Hausdorff mea- sure at dimensions0. This measure in turn induces a measure on the space of choices. Given a choice, the commutator of D with a Lipschitz continuous function can be interpreted as a directional derivative. By integrating over all choices, this leads to the definition of an analogue of the Laplace-Beltrami operator. This operator has compact resolvent and generates a Markov semigroup which plays the role of a Brownian motion on C. This construction is applied to the simplest case, the triadic Cantor set, where: (i) the spectrum and the eigenfunctions of the Laplace-Beltrami operator are computed, (ii) the Weyl asymptotic formula is shown to hold with the dimensions0, (iii) the corresponding Markov process is shown to have an anomalous diffusion with E.d.Xt;Xt Cit/ 2 / 'it ln.1=it/ asit #0.