The paper deals with Gibbs samplers that include high-dimensional conditional Gaussian distributions. It proposes an efficient algorithm that only requires a scalar Gaussian sampling. The algorithm relies on a random excursion along a random direction. It is proved to converge, i.e. the drawn samples are asymptotically under the target distribution. Our original motivation is in unsupervised inverse problems related to general linear observation models and their solution in a hierarchical Bayesian framework implemented through sampling algorithms. The paper provides an illustration focused on 2-D simulations and on the super-resolution problem.