Weiss' method of quantization of field theories characterized by first-order Lagrangians can be carried out in a non-metrical "amorphous" space, as was first stated by Bergmann and Brunings. The gravitational equations can be regarded as differential equations for the field variables ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$ in an amorphous space and the quantization procedure can be applied to them. The gravitational field equations are written in canonical form, the Hamiltonian being a function of generalized coordinates, momenta, and velocities. This Hamiltonian is obtained using a method developed by Dirac for Lorentz invariant theories.