Holmboe waves are long-lived traveling waves commonly found in environmental stratified shear flows in which a relatively sharp, stable density interface is embedded within a more diffuse shear layer. Although previous research has focused on their linear properties (the Holmboe instability), and on their turbulent properties (Holmboe wave turbulence), little is known about their finite-amplitude properties in the nonlinear but non-turbulent regime. In this paper we tackle this problem with a weakly-nonlinear temporal stability analysis of Holmboe waves.