Below the freezing temperature ${\mathit{T}}_{\mathit{g}}$, the configuration space of spin glasses is characterized by the existence of quasidegenerate equilibrium states whose number increases drastically as the temperature is lowered. The observed time dependence of the dynamical properties (aging effects) can be understood as the trend of the system towards thermal equilibrium between all accessible states. We present a series of temperature cycling experiments on the time decay of the remanent magnetization in Ag:Mn(2.6 at. %). The results are consistent with the picture of an ultrametric organization of the metastable states, predicted by the Parisi solution of the Sherrington-Kirkpatrick model. Within this interpretation, the temperature dependence of the height of the barrier between two states is determined experimentally. We also establish a quantitative relationship between the barrier height ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{\beta}}}$ and the Hamming distance ${\mathit{d}}_{\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{\beta}}}$ separating two metastable states \ensuremath{\alpha} and \ensuremath{\beta}. This experimental result is compared with numerical simulations on mean-field spin glasses.