  We define and study the motive of the moduli stack of vector bundles of fixed
rank and degree over a smooth projective curve in Voevodsky's category of
motives. We prove that this motive can be written as a homotopy colimit of
motives of smooth projective Quot schemes of torsion quotients of sums of line
bundles on the curve. When working with rational coefficients, we prove that
the motive of the stack of bundles lies in the localising tensor subcategory
generated by the motive of the curve, using Bialynicki-Birula decompositions of
these Quot schemes. We conjecture a formula for the motive of this stack, and
we prove this conjecture modulo a conjecture on the intersection theory of the
Quot schemes.
