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Using three-dimensional numerical simulations, we have found that the weak-field magnetorotational instability constitutes a hydromagnetic dynamo in astrophysical disks. The field amplification mechanism lies outside the scope of kinematic dynamo theory and implies that kinematic dynamo theory is not applicable to accretion disk systems. <P />We begin with simulations of isotropic, homogeneous turbulence to demonstrate the ability of our finite-difference code to reproduce known dynamos. To calibrate our code, we measure magnetic field rates when the turbulent velocity field is not sustained through external forcing. When the turbulence is maintained with imposed velocity fields, a magnetic dynamo can be produced. Two cases are considered: velocity fields with and without net helicity. Both the helical and the nonhelical velocity fields amplify and sustain magnetic fields despite substantial turbulent dissipation. The nonhelical dynamo increases the magnetic field energy by an order of magnitude. The helical dynamo produces greater amplification (a factor of 30 in energy) and a more ordered field. Lorentz forces eventually limit the growth of the magnetic field in both cases. <P />We next carry out simulations in the local shearing box model of an accretion disk. The initial magnetic field has a mean field value of zero over the computational domain and is weak in the sense that the initial Alfvén speed is much less than the sound speed. The magnetorotational instability rapidly generates turbulence within the disk. The turbulence is anisotropic, producing significant Maxwell and Reynolds stresses that transport angular momentum outward. Both the stresses and magnetic energy density remain subthermal and do not approach equipartition. The magnetic energy is amplified by a factor of 20 and maintained for over 200 orbits of time, far longer than the magnetic decay time in the absence of an amplification mechanism. A parameter survey indicates the final turbulent state does not depend on the initial average field strength; some dependence on the computational domain size, magnetic Prandtl number, and numerical resolution is observed. Finally, we demonstrate explicitly that more than shear and turbulence are required to produce dynamo amplification of magnetic fields.