In　recent　years,　it　has　been　found　that　many　phenomena　in　engineering,　physics,　chemistry　and　other　sciences　can　be　described　very　successfully　by　models　using　mathematical　tools　from　Fractional　Calculus.　Recently,　a　new　space　and　time　fractional　Bloch-Torrey　equation　(ST-FBTE)　has　been　proposed　(Magin　et　al.,　J.　Magn.　Reson.　190(2),　255-270,　2008),　and　successfully　applied　to　analyse　diffusion　images　of　human　brain　tissues　to　provide　new　insights　for　further　investigations　of　tissue　structures.　In　this　paper,　we　consider　the　ST-FBTE　with　a　nonlinear　source　term　on　a　finite　domain　in　three-dimensions.　The　time　and　space　derivatives　in　the　ST-FBTE　are　replaced　by　the　Caputo　and　the　sequential　Riesz　fractional　derivatives,　respectively.　Firstly,　we　propose　a　spatially　second-order　accurate　implicit　numerical　method　(INM)　for　the　ST-FBTE　whereby　we　discretize　the　Riesz　fractional　derivative　using　a　fractional　centered　difference.　Secondly,　we　prove　that　the　implicit　numerical　method　for　the　ST-FBTE　is　uniquely　solvable,　unconditionally　stable　and　convergent,　and　the　order　of　convergence　of　the　implicit　numerical　method　is　O　(τ2-α+τ+hx2+hy2}+hz2).　Finally,　some　numerical　results　are　presented　to　support　our　theoretical　analysis.　©　2013　Springer　Science+Business　Media　New　York.