In this paper, we consider a cognitive satellite-terrestrial network where the satellite communication operates in the microwave frequency bands allocated to terrestrial networks in an underlay mode. Taking the statistical delay quality-of-service (QoS) requirements into account, we investigate the effective capacity of the satellite network while satisfying interference-power limitations imposed by terrestrial networks. Specifically, the primary terrestrial transmitters that would result in aggregate interference at the satellite receiver are modeled as points of a Poisson point process (PPP). By characterizing the aggregate interference as a gamma distribution, we obtain a closed-form expression for the effective capacity of the secondary satellite network. Finally, simulation results are provided to not only demonstrate the validity of the theoretical results, but also show the effects of system parameters such as delay exponent of satellite communications, interference-power limitations of terrestrial networks, and intensity of terrestrial transmitters on the performance of the satellite network.