In　this　article　we　consider　a　two-level　finite　element　Galerkin　method　using　mixed　finite　elements　for　the　two-dimensional　nonstationary　incompressible　Navier-Stokes　equations.　The　method　yields　a　H-1-optimal　velocity　approximation　and　a　L-2-optimal　pressure　approximation.　The　two-level　finite　element　Galerkin　method　involves　solving　one　small,　nonlinear　Navier-Stokes　problem　on　the　coarse　mesh　with　mesh　size　H,　one　linear　Stokes　problem　on　the　fine　mesh　with　mesh　size　h　much　less　than　H.　The　algorithm　we　study　produces　an　approximate　solution　with　the　optimal,　asymptotic　in　h,　accuracy.