We developed a new soft x-ray speckle intensity correlation spectroscopy system by use of a single shot high brilliant plasma soft x-ray laser. The plasma soft x-ray laser is characterized by several picoseconds in pulse width, more than 90% special coherence, and 10(11) soft x-ray photons within a single pulse. We developed a Michelson type delay pulse generator using a soft x-ray beam splitter to measure the intensity correlation of x-ray speckles from materials and succeeded in generating double coherent x-ray pulses with picosecond delay times. Moreover, we employed a high-speed soft x-ray streak camera for the picosecond time-resolved measurement of x-ray speckles caused by double coherent x-ray pulse illumination. We performed the x-ray speckle intensity correlation measurements for probing the relaxation phenomena of polarizations in polarization clusters in the paraelectric phase of the ferroelectric material BaTiO(3) near its Curie temperature and verified its performance.