The nearly rotationally resolved spectrum of the A (2)A(')<--X (2)A(") 0(0)(0) transition of perdeutero methyl peroxy near 1.35 microm has been studied via pulsed cavity ringdown spectroscopy. Albeit, this is a weak transition, it is possible to observe the spectrum under jet-cooled conditions (approximately 15 K) by combining a source of narrow-bandwidth radiation (approximately 250 MHz) with a supersonic slit-jet expansion incorporating an electric discharge. The near infrared radiation was obtained by using stimulated Raman scattering and a pulsed, nearly Fourier-transform-limited Ti:sapphire amplifier seeded by a scanable cw Ti:sapphire ring laser. The experimental spectrum has been fitted using a model Hamiltonian that includes the rigid body rotation of an asymmetric top and the spin-rotation interaction. An excellent quality fit was obtained resulting in the determination of 15 molecular parameters characterizing the A and X states. Other results reported for CD(3)O(2) include an estimate of the radical concentration and the vibronic transition dipole from the observed absorption intensities. Details about the spectral linewidths are also discussed.