The oxidation by oxygen of a zirconium based alloy, M5(TM) (which is a ZrNbO alloy, containing 1% of Nb) has been studied. The M5(TM) alloy, like many zirconium alloys, undergoes a kinetic transition. The aim of the present work is to achieve a better understanding of the oxidation in the pre-transition stage, and to clearly identify the differences between the pre- and post transition stages from the kinetic point of view. The oxidation of M5(TM) was followed by isothermal gravimetry at 490degreesC, under a controlled partial pressure of oxygen (in the range 7 to 200hPa). First, we have verified the steady state assumption, by coupling thermogravimetry and differential scanning calorimetry (DSC) : it is shown that the system is in a steady state from the beginning of the oxidation, in the pre- and post-transition stages. Then, the existence of a rate-limiting step was verified in the pre-transition stage using an experimental method based on temperature or pressure jumps; this assumption is no more verified in the post-transition stage, which means that the oxidation does not proceed in the same way as in the pre-transition stage. Finally, we have obtained the variations of zirconia growth reactivity Phi with the oxygen pressure, in the pre-transition stage (using the pressure jump method). The oxygen pressure has a slightly accelerating effect, which cannot be interpreted by the diffusion of oxygen vacancies through a dense oxide layer (in that case no effect of the oxygen pressure would be observed). In the post-transition stage, the oxygen effect is more important.