  We establish a sharp bilinear estimate for the Klein--Gordon propagator in
the spirit of recent work of Beltran--Vega. Our approach is inspired by work in
the setting of the wave equation due to Bez, Jeavons and Ozawa. As a
consequence of our main bilinear estimate, we deduce several sharp estimates of
null-form type and recover some sharp Strichartz estimates found by Quilodr\'an
and Jeavons.
