Conjugate prior of a Laplace distribution

Does anyone know what the Bayesian conjugate prior of a Laplace distribution likelihood is?

Most of the exponential family of distributions have known conjugate priors but the Laplace is not in the exponential family in general. I've been struggling for a while trying to find information on its conjugate prior but I haven't found anything.

I'm working on a model that would be greatly simplified if I could produce the conjugate for the Laplacian. I am thinking it might be related to the conjugate of the (single-) exponential.