A cooling arrangement for a gas turbine engine includes: (a) a turbine nozzle having: (i) spaced-apart arcuate inner and outer bands; and (ii) a hollow, airfoil-shaped turbine vane extending between the inner and outer bands, the vane disposed in a primary flowpath of the engine; (b) a supporting structure coupled to the outer band such that an outer band cavity is defined between the outer band and the stationary structure; (c) a first conduit passing through the outer band cavity and communicating with the interior of the vane, the first conduit coupled to a first source of cooling air within the engine; and (d) a second conduit communicating with the outer band cavity, the second conduit coupled to a second source of cooling air within the engine.