In the periodic three-dimensional Navier-Stokes equations \(\partial _tu+u\cdot \nabla u-\nu \Delta u=-\nabla p, div u=0, u_{t=0}=u_0\) the initial data \(u_0\) is decomposed as \(u_0=v_0+w_0\), where \(w_0\) does not depend on the variable \(x_3 (u=u(x_1,x_2,x_3,t))\). The author proves that if \(\left\|w_0\right\|_X\exp (\left\|v_0\right\|_Y^2/C\nu ^2)\leq C\nu \), then global existence and uniqueness for the Navier-Stokes equations hold. The spaces \(X\) and \(Y\) have different structures w.r.t. the variables \((x_1,x_2)\) and \(x_3\). Their components are analogous to \(L_2\), to Besov's spaces and to a space with the norm expressed in terms of the Fourier coefficients in \(l^2\) with weight.