We present a NLO study of inclusive polarized prompt photon production in a conceivable fixed target pp mode of HERA with longitudinally polarized protons at √ s = 39 GeV.We analyze the sensitivity of the corresponding double spin asymmetry to the proton's polarized gluon distribution ∆g and estimate the expected statistical precision in its determination. The main theoretical uncertainties in the predictions are examined.It has recently become possible to perform a complete and consistent study of longitudinally polarized deep-inelastic scattering (DIS) in next-to-leading order (NLO) of QCD, since the spin-dependent two-loop splitting functions, needed for the NLO evolution of the polarized parton distributions, have become available [1,2]. A first such phenomenological NLO analysis, taking into account all available experimental data on polarized DIS [3] has been presented in [4], followed by the analyses [5,6]. The studies of [4,6] have shown that present polarized DIS data are still quite far from providing accurate knowledge about the nucleon's spin-dependent sea quark and gluon distributions. This holds true, in particular, for the polarized gluon density ∆g, the x-shape of which seems to be hardly constrained at all [4,6] by the DIS data, even though a tendency towards a sizeable positive total gluon polarization, 1 0 ∆g(x, Q 2 = 4 GeV 2 )dx 1, was found [4,5,6]. Thus, there is clearly some need for independent information on ∆g. For this purpose, it seems expedient to look at processes for which ∆g enters in leading order (LO) already, rather than as a NLO correction as for the spin-dependent DIS structure function g 1 . One of such processes is inclusive large-p T prompt photon production in collisions of longitudinally polarized protons, p p → γX [7,8,9,10]. In the unpolarized case where this process has been studied in a huge number of experiments it has been an invaluable tool for pinning down the proton's unpolarized gluon distribution g(x, Q 2 ) [11,12,13,14]. Hence prompt photon production with polarized beams seems a promising source for obtaining information on ∆g.It is being discussed as one future option for HERA to polarize its 820 GeV proton beam. If this can be achieved, one could use the beam in a fixed target experiment, scattering it off an internal polarized nucleon target. This conceivable constellation, dubbed , would yield √ s ≈ 39 GeV and thus could provide information complementary to that obtained from planned similar spin physics experiments at much higher energies at the RHIC collider [16]. Theoretical predictions for polarized prompt photon production at √ s ≈ 40 GeV have been made in the past [8,10], taking into account the spin-dependent 'direct' subprocess cross sections for ab → γX (a, b = q,q, g) including their full NLO QCD corrections as calculated in [8,9]. From the experience in the unpolarized case, the inclusion of NLO corrections is expected to be quite important 1 in order to make reliable predictions. The main shortcoming of the studies [8,10] was, however...