We synthesized tetragonal a-FeSe by melting a powder mixture of iron and selenium at high pressure. Subsequent annealing at normal pressure results in removing traces of hexagonal bFeSe, formation of a rather sharp transition to superconducting state at T c ~ 7 K, and the appearance of a magnetic transition near T M = 120 K. Resistivity and ac-susceptibility were measured on the annealed sample at hydrostatic pressure up to 4.5 GPa. A magnetic transition visible in ac-susceptibility shifts down under pressure and the resistive anomaly typical for a spin density wave (SDW) antiferromagnetic transition develops near the susceptibility anomaly. T c determined by the appearance of a diamagnetic response in susceptibility, increases linearly under pressure at a rate dT c /dP = 3.5 K/GPa. Below 1.5 GPa, the resistive superconducting transition is sharp; the width of transition does not change with pressure; and, T c determined by a peak in dr/dT increases at a rate ~ 3.5 K/GPa. At higher pressure, a giant broadening of the resistive transition develops. This effect cannot be explained by possible pressure gradients in the sample and is inherent to a-FeSe. The dependences dr(T)/dT show a signature for a second peak above 3 GPa which is indicative of the appearance of another superconducting state in a-FeSe at high pressure. We argue that this second superconducting phase coexists with SDW antiferromagnetism in a partial volume fraction and originates from pairing of charge carriers from other sheets of the Fermi surface.2 Introduction Recent discovery of high T c superconductivity in layered iron arsenides [1][2][3][4] extended the family of unconventional superconductors. Previous work on high T c cuprates and heavy fermion superconductors prepared a basis for rapid progress in research on these new materials. The proximity to a magnetic instability, complex gap function, and coexistence of magnetism and superconductivity provide a key framework for future understanding the pairing mechanism of iron-based superconductors. The common structural feature of all these materials is layers composed of edge-sharing FeAs 4 -tetrahedra, separated by rare-earth-oxygen or alkaline-earth layers. The tetragonal a-phase of iron selenide has a structure composed of a stack of edge-sharing FeSe 4 -tetrahedra layer by layer, without any additional separating elements and may be regarded as an end member of a series of ironbased superconductors. Therefore, the discovery of superconductivity in a-FeSe with T c = 7 K [5] attracted considerable attention. Specific heat [5] and NMR [6] measurements indicate the unconventional nature of superconductivity in a-FeSe with lines of vanishing gap on the Fermi surface. In spite of the relatively simple structure of this binary compound, the preparation of a single phase sample is a challenge. Standard solid state reaction usually results in two or more phases in the sample, the impurity hexagonal b-phase being dominant [5][6][7][8][9]. In the first publications on superconductivity in iron seleni...