The standard magnetorotational instability (SMRI) has been regarded as the most promising instability responsible for the turbulence required to explain the fast accretion observed across the Universe. However, unlike other fundamental plasma processes such as Alfvén waves and magnetic reconnection which have been subsequently detected and studied in space and in the laboratory, SMRI remains unconfirmed even for its existence long after its proposal, despite its widespread applications in modelling including recent black hole imaging. Its direct detection has been hindered in observations due to its microscopic nature at astronomical distances, and in the laboratory due to stringent requirements and interferences from other processes. Here we report the first direct evidence showing that SMRI indeed exists in a novel laboratory setup where a uniform magnetic field is imposed along the axis of a differentially rotating flow of liquid metal confined radially between concentric cylinders and axially by copper end-rings. Through in situ measurement of the radial magnetic field B_r at the inner cylinder, onset of the axisymmetric SMRI is identified from the nonlinear increase of B_r beyond a critical magnetic Reynolds number. Further analysis reveals that the SMRI is accompanied by a nonaxisymmetric m=1 mode, which is a linear instability having an exponential growth at its onset. Further analysis excludes the possibility that the m=1 mode is the conventional Rayleigh instability or the Stewartson-Shercliff layer instability, implying that it could be a non-axisymmetric version of SMRI that breaks the rotational symmetry of the system. The experimental results are reproduced by nonlinear three-dimensional numerical simulations, which further show that SMRI causes the velocity and magnetic fields to contribute an outward flux of axial angular momentum in the bulk region, just as it does in accretion disks