Organic semiconductors that are pi-conjugated are emerging as an important platform for 'spintronics', which purports to harness the spin degree of freedom of a charge carrier to store, process and/or communicate information(1). Here, we report the study of an organic nanowire spin valve device, 50 nm in diameter, consisting of a trilayer of ferromagnetic cobalt, an organic, Alq(3), and ferromagnetic nickel. The measured spin relaxation time in the organic is found to be exceptionally long - between a few milliseconds and a second - and it is relatively temperature independent up to 100 K. Our experimental observations strongly suggest that the primary spin relaxation mechanism in the organic is the Elliott - Yafet mode, in which the spin relaxes whenever a carrier scatters and its velocity changes.