Let M° be a complete noncompact manifold and g an asymptotically conic Riemaniann metric on M°, in the sense that M° compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M. Let Δ be the positive Laplacian associated to g, and P = Δ+V, where V is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure dE(λ) = (λ/πi)(R(λ + i0) (R(λ - i0)) of P1/2 +, where R(λ) = (P - λ 2)-1, as λ → 0, in a manner similar to that done by the second author and Vasy (2001) and by the first two authors (2008, 2009). The main result is that the spectral measure has a simple, 'conormal- Legendrian' singularity structure on a space which was introduced in the 2008 work of the first two authors and is obtained from M2 ×[0, λ0) by blowing up a certain number of boundary faces. We use this to deduce results about the asymptotics of the wave solution operators cos(t√P+) and sin(t√P+)/√P+, and the Schrödinger propagator eitP+, as t→∞. In particular, we prove the analogue of Price's law for odd-dimensional asymptotically conic manifolds.