Cezve doesn't leak with cold water in it, leaks after heating
You would have to find the crack (presumably there is a crack which is tight when cold, but opens with heating due to thermal expansion) and then you would need to fix it, or have someone else fix it, with a food safe (lead and cadmium free) solder or brazing material. Whether this is economical or not depends on many factors:


Can you do it yourself, do you know somone who can, or will you need to hire a professional
Are professionals who do "pot and pan repair" easily found in your area or not? 
How much does a new coffeepot (cezve) cost relative to the cost of getting it fixed?