The inference of gene association networks from gene expression profiles is an important approach to elucidate various cellular mechanisms. However, there exists a problematic issue that the number of samples is relatively small than that of genes. A promising approach to this problem will be to design regularization terms for characteristic network structures like sparsity and scale-freeness and optimize a scoring function including those regularization terms. The inference problem for gene association networks is often formulated as the problem of estimating the inverse covariance matrix of a Gaussian distribution from its samples. For this Bayesian inference problem, we propose a novel scale-free structure prior and devise a sampling method for optimizing a posterior probability including the prior. In a simulation study, scale-free graphs of 30 and 100 nodes are generated by the Barabási-Albert model, and the proposed method is shown to outperform another method which also use a scale-free regularization term. Our method is also applied to real gene expression profiles, and the resulting graph shows biologically meaningful features. Thus, we empirically conclude that our scale-free structure prior is effective in Bayesian inference of Gaussian graphical models.