The current popular method for approximate simulation from the posterior distribution of the linear Bayesian LASSO is a Gibbs sampler. It is well-known that the output analysis of an MCMC sampler is difficult due to the complex dependence amongst the states of the underlying Markov chain. Practitioners can usually only assess the convergence of MCMC samplers using heuristics. In this paper we construct a method that yields an independent and identically distributed (iid) draws from the LASSO posterior. The advantage of such exact sampling over the MCMC sampling is that there are no difficulties with the output analysis of the exact sampler, because all the simulated states are independent. The proposed sampler works well when the dimension of the parameter space is not too large, and when it is too large to permit exact sampling, the proposed method can still be used to construct an approximate MCMC sampler.