One-dimensional topological superconductors harbor Majorana bound states at their ends. For superconducting wires of finite length L, these Majorana states combine into fermionic excitations with an energy ε₀ that is exponentially small in L. Weak disorder leaves the energy splitting exponentially small, but affects its typical value and causes large sample-to-sample fluctuations. We show that the probability distribution of ε₀ is log normal in the limit of large L, whereas the distribution of the lowest-lying bulk energy level ε₁ has an algebraic tail at small ε₁. Our findings have implications for the speed at which a topological quantum computer can be operated.