Cost of one-relator groups

For any infinite one-relator group $\mathrm{\Gamma }=⟨S\mid w^m⟩$, we prove that $\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}(\mathrm{\Gamma })=\mathrm{\mid }S\mathrm{\mid }-\frac{1}{m}$. For such groups, this gives ${\beta }_1^{(2)}(\mathrm{\Gamma })=\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}(\mathrm{\Gamma })-1$, answering a special case of Gaboriau’s question on the relationship between cost and first ${\mathrm{\ell }}^2$-Betti number.