The conversion of silicon-based polymers into a ceramic occurs by the release of small organic molecules at intermediate temperatures, and hydrogen at the higher temperatures. The conversion is accompanied by significant densification. In order for the body to emerge without fracture damage, the shrinkage must be accommodated by viscous flow. In this paper we show that the viscosity changes during the conversion, going through a sharp minimum near 600 degrees C. In the regime of hydrogen evolution, which begins near 700 degrees C, fairly steady viscous flow leads to large shrinkage. The viscosity increases rapidly as 1000 degrees C is approached, pointing to the exhaustion of hydrogen. The results presented here may be used to optimize the time-temperature protocol for achieving damage-free specimens: heating quickly up to similar to 700 degrees C and then slowly at higher temperatures to allow time for viscous relaxation.