The classical notion of the Darboux transformation of isothermic surfaces can be generalised to a transformation for conformal immersions. Since a minimal surface is Willmore, we can use the associated mathbb { C}_*-family of flat connections of the harmonic conformal Gauss map to construct such transforms, the so-called mu -Darboux transforms. We show that a mu -Darboux transform of a minimal surface is not minimal but a Willmore surface in 4-space. More precisely, we show that a mu -Darboux transform of a minimal surface f is a twistor projection of a holomorphic curve in mathbb { C}mathbb { P}^3 which is canonically associated to a minimal surface f_{p,q} in the right-associated family of f. Here we use an extension of the notion of the associated family f_{p,q} of a minimal surface to allow quaternionic parameters. We prove that the pointwise limit of Darboux transforms of f is the associated Willmore surface of f at mu =1. Moreover, the family of Willmore surfaces mu -Darboux transforms, mu in mathbb { C}_*, extends to a mathbb { C}mathbb { P}^1 family of Willmore surfaces f^mu : M rightarrow S^4 where mu in mathbb { C}mathbb { P}^1.