Single crystal inelastic neutron scattering is used to study dynamic spin correlations in the quasi-one-dimensional quantum antiferromagnet BaCu2(Si0.5Ge0.5)2O7, where the exchange constant fluctuates due to a random distribution of Si and Ge atoms. The measured low-energy spectrum is dominated by localized excitations and can be understood in the framework of the random singlet model. The observed scaling relations for the frequency dependencies of the correlation length and structure factor are in excellent agreement with recent theoretical predictions for the renormalization group fixed point.