Liquid freezing is a crucial issue in many different science fields, from aircraft industry to geophysical flows (lava flows, waterfall ice formation...) that remains a very large question to be addressed. Although there is research on the solidification of liquid at rest (e.g droplet freezing), there are still few studies on the freezing of capillary flows. Here we propose an experimental study of the competition between the flow of a water rivulet and its solidification. In our set up, water flows down from a capillary tube to a metal inclined plane which is cooled to subzero temperatures. A UV-sensitive fluorescent tracer, fluorescein, is diluted at low concentration into water in or- der to better track the solidification front during ice accretion. Using this set up, several experimental parameters have been varied such as water flow rate, water entry temperature and metal temperature. Main results shows that, after a typical time, the system reaches a stationary state where water flows on a static three dimensional complex ice structure. This structure presents a sur- prising linear increase of its thickness with distance from the water injection point. Furthermore, it appears that water retracts on its ice forming a non-zero contact angle, making three dimensional study of the ice structure particularly interesting. The mechanism proposed to explain such structure is a competition between capillary hydrodynamics, and solidification.