  We present a study of the application of the Jagiellonian Positron Emission
Tomograph (J-PET) for the registration of gamma quanta from decays of
ortho-positronium (o-Ps). The J-PET is the first positron emission tomography
scanner based on organic scintillators in contrast to all current PET scanners
based on inorganic crystals. Monte Carlo simulations show that the J-PET as an
axially symmetric and high acceptance scanner can be used as a multi-purpose
detector well suited to pursue research including e.g. tests of discrete
symmetries in decays of ortho-positronium in addition to the medical imaging.
The gamma quanta originating from o-Ps decay interact in the plastic
scintillators predominantly via the Compton effect, making the direct
measurement of their energy impossible. Nevertheless, it is shown in this paper
that the J-PET scanner will enable studies of the o-Ps$\to3\gamma$ decays with
angular and energy resolution equal to $\sigma(\theta) \approx 0.4^{\circ}$ and
$\sigma(E) \approx 4.1$ keV, respectively. An order of magnitude shorter decay
time of signals from plastic scintillators with respect to the inorganic
crystals results not only in better timing properties crucial for the reduction
of physical and instrumental background, but also suppresses significantly the
pileups, thus enabling compensation of the lower efficiency of the plastic
scintillators by performing measurements with higher positron source
activities.
