A new approach to cognitive radio is presented.  The cognitive transmitter/receiver pair is deployed in an overlay fashion  with a pre-existing primary user, coded or uncoded.  The basic idea of the new approach is to exploit the fact that, typically the primary channel is not fully loaded by the primary user service, thus leaving a non-negligible margin to accommodate the cognitive transmission. Furthermore, the exploitation of such a load margin is optimized by shaping the spectrum of the transmitted cognitive signal. It is shown that non-trivial coexistence of primary and cognitive systems is possible even without sharing the primary user message at the cognitive transmit side. Surprisingly, the cognitive transmitter can be designed such that it is no longer limited by its interference power at the primary receiver, and can always transmit at its full available device power. Analytical development and numerical illustrations are presented, in particular focusing on the logarithmic growth rate, i.e., the prelog coefficient, of cognitive transmission in the high-power regime.  For coded primary users, surprisingly, the optimal pre-log coefficient of the cognitive user rate is a function only of the parameters of the primary user channel.