A detector using liquid Xenon (LXe) in the scintillation mode is studied for Positron Emission Tomography (PET) of small animals. Its specific design aims at taking full advantage of the Liquid Xenon scintillation properties. This paper reports on energy, time and spatial resolution capabilities of the first LXe prototype module equipped with a Position Sensitive Photo- Multiplier tube (PSPMT) operating in the VUV range (178 nm) and at 165 K. The experimental results show that such a LXe PET configuration might be a promising solution insensitive to any parallax effect.