Conversion between signals in the microwave and optical domains is of great interest both for classical telecommunication and for connecting future superconducting quantum computers into a global quantum network. For quantum applications, the conversion has to be efficient, as well as operate in a regime of minimal added classical noise. While efficient conversion has been demonstrated using mechanical transducers, they have so far all operated with a substantial thermal noise background. Here, we overcome this limitation and demonstrate coherent conversion between gigahertz microwave signals and the optical telecom band with a thermal background of less than one phonon. We use an integrated, on-chip electro-optomechanical device that couples surface acoustic waves driven by a resonant microwave signal to an optomechanical crystal featuring a 2.7 GHz mechanical mode. We initialize the mechanical mode in its quantum ground state, which allows us to perform the transduction process with minimal added thermal noise, while maintaining an optomechanical cooperativity >1, so that microwave photons mapped into the mechanical resonator are effectively upconverted to the optical domain. We further verify the preservation of the coherence of the microwave signal throughout the transduction process.