Turbulent flow of superfluid (4)He at mK temperatures around an oscillating microsphere is known to be unstable at low driving forces, switching intermittently between turbulent and laminar phases. The lifetimes of the turbulent phases are exponentially distributed, and the mean lifetimes grow exponentially with the square of the driving force. These experimental results are attributed to statistical fluctuations of the density L of the vortex line length. As a result, a normal probability distribution of L(2) is found having a standard deviation of 2.9x10(14) m(-4) and a spectral bandwidth Deltaomegaapproximate to13 s(-1).