Many　physical　processes　appear　to　exhibit　fractional　order　behavior　that　may　vary　with　time　and/or　space.　The　continuum　of　order　in　the　fractional　calculus　allows　the　order　of　the　fractional　operator　to　be　considered　as　a　variable.　In　this　paper,　we　consider　a　new　space-time　variable　fractional　order　advection-dispersion　equation　on　a　finite　domain.　The　equation　is　obtained　from　the　standard　advection-dispersion　equation　by　replacing　the　first-order　time　derivative　by　Coimbra＇s　variable　fractional　derivative　of　order　alpha(X)　is　an　element　of　(0,1],　and　the　first-order　and　second-order　space　derivatives　by　the　Riemann-Liouville　derivatives　of　order　gamma(X,　t)　is　an　element　of(0,　1]　and　beta(X,　t)　is　an　element　of(1,2],　respectively.　We　propose　an　implicit　Euler　approximation　for　the　equation　and　investigate　the　stability　and　convergence　of　the　approximation.　Finally,　numerical　examples　are　provided　to　show　that　the　implicit　Euler　approximation　is　computationally　efficient.　(C)　2014　Elsevier　Inc.　All　rights　reserved.