An inbred sunflower line, designated H1063R, is disclosed. The invention relates to the seeds of inbred sunflower line H1063R, to the plants of inbred sunflower line H1063R and to methods for producing a sunflower plant, either inbred or hybrid, by crossing the inbred line H1063R with itself or another sunflower line. The invention further relates to methods for producing a sunflower plant containing in its genetic material one or more transgenes and to the transgenic plants produced by that method and to methods for producing other inbred sunflower lines derived from the inbred H1063R.