A new three-time-level explicit economical scheme (acronym 3TL-EEC) has been proposed that allows a stable time step for gravity-wave propagation that is twice as large as the stable time step permitted by the standard three-time-level leapfrog scheme. In a two-time-level frame-work, the well-established and widely-used forward-backward scheme is already known to allow a stable time step for gravity-wave propagation that is twice the time step allowed by the leapfrog scheme. However, when the forward-backward scheme is applied over two time steps (say, from the time-level n-1 to n+1) of the leapfrog scheme, the computational economy of the forward-backward scheme over the leapfrog scheme is lost. Thus, there is no benefit in computational efficiency if the forward-backward scheme is applied for the gravity waves in combination with the leapfrog scheme, e.g., for the advection and Coriolis terms. The proposed 3TL-EEC scheme was derived from the forward-backward scheme using an original and innovative approach. The new scheme recaptures the latter scheme's attractive features of computational economy in a three-time-level frame-work, and thereby allows for the use of the leapfrog scheme for advection and Coriolis effects. Detailed stability properties of the 3TL-EEC scheme will be presented for the 1D shallow-water (pure) gravity waves and 2D shallow-water gravity waves on an f-plane with uniform advection; and compared with the corresponding properties of the leapfrog scheme. The 3TL-EEC scheme, in conjunction with the leapfrog scheme for advection and Coriolis effects, has been implemented in a global grid-point shallow-water model that conserves potential enstrophy and total energy on the Arakawa C grid. Lastly, the numerical accuracy and stability of the proposed scheme will be demonstrated and compared to the same aspects of the leapfrog scheme, via numerical time integrations of the aforementioned global shallow-water model, starting with the Rossby-Haurwitz wave number 4 initial conditions.