We consider the problem of steering a linear dynamical system with complete state observation from an initial Gaussian distribution in state-space to a final one with minimum energy control. The system is stochastically driven through the control channels; an example for such a system is that of an inertial particle experiencing random "white noise" forcing. We show that a target probability distribution can always be achieved in finite time. The optimal control is given in state-feedback form and is computed explicitly by solving a pair of differential Lyapunov equations that are nonlinearly coupled through their boundary values. This result, given its attractive algorithmic nature, appears to have several potential applications such as to quality control, control of industrial processes, as well as to active control of nanomechanical systems and molecular cooling. The problem to steer a diffusion process between end-point marginals has a long history (Schrödinger bridges) and the present case of steering a linear stochastic system constitutes such a Schrödinger bridge for possibly degenerate diffusions. Our results provide the first implementable form of the optimal control for a general Gauss-Markov process. Illustrative examples are provided for steering inertial particles and for "cooling" a stochastic oscillator. A final result establishes directly the property of Schrödinger bridges as the most likely random evolution between given marginals to the present context of linear stochastic systems. A second part to this work, that is to appear as part II, addresses the general situation where the stochastic excitation enters through channels that may differ from those used to control.