Let \(D\subset\mathbb R^2\) be a bounded, simply connected domain with smooth boundary and let \(\omega_j\) (\(j=1,\dots ,n\)) be disjoint, smooth, simply connected open sets contained in \(D\). Set \(\Omega=D\setminus\cup_{j=1}^n\omega_j\). This paper is devoted to the asymptotic study as \(\varepsilon\rightarrow 0\) of the Ginzburg-Landau free energy \(E_\varepsilon (\psi ,A)=\int_\Omega [2^{-1}| (\nabla -iA)\psi | ^2+(4\varepsilon^2)^{-1}(| \psi| ^2-1)^2]dx +{2}^{-1}\int_D(h-h_{ex})^2dx\). One of the most important consequences of the energy on a domain with holes is that there is a rich class of “vortexless” configurations, obtained as critical points of \(E_\varepsilon\) subject to the constraint \(| u| =1\): \[ E_\infty (u ,A)=\int_\Omega 2^{-1}| (\nabla -iA)u| ^2dx+ +{2}^{-1}\int_D(h-h_{ex})^2dx, \] where \(u\in H^1(\Omega ; \mathbb S^1)\) and \(A\in H^1(D;\mathbb R^2)\). After recalling some basic facts related to the minimization of the Ginzburg-Landau energy with magnetic field, the authors study the minimization problem associated to \(E_\infty\) as a function of the applied field \(h_{ex}\). Next, it is showed that minimizers of the limit problem are completely characterized by the magnetic field \(h=\)curl\(\,A\), which behaves like \(h_{ex}(1-\zeta (x))\), where \(\zeta\) solves the Dirichlet problem \(-\Delta\zeta +\zeta =1\) in \(\Omega\), \(\zeta=0\) on \(\partial\Omega\). The proofs are based on elliptic estimates and adequate variational methods.