If, when the laser is not pulsed, the light through the sample is a fixed level (i.e. there is no external modulation on it by means of a chopper) then it appears that there will be a signal that is repetitive and triggered by the laser when it is pulsed. Given this, you can use a signal averager running at the laser rep rate of 10Hz to record many waveforms and average them.
For this you can use the model 9846 transient recorder (as the repetition rate is low) or, if you might want to move to higher repetition rates in the future, the model 9826 or Eclipse (the latter only if your 10 Hz laser can be externally triggered). Your existing scope may even be able to do the averaging.
If the "normal" light intensity is drifting with time during the experiment then the above technique will not work well. Assuming that your laser gives a (stable) 10 Hz trigger output then I would use the following system:
Use the laser to trigger a model 9650A delay generator, with the delay set to 99.8 ms. Take the trigger out of this and use it trigger a 9846, set to 32 ns per point and 65 k points and a sweep time of 2 ms. Hence on each 100 ms trigger of the laser, the 9846 will acquire a sweep of data "bracketing" the laser pulse. Each sweep is then read by the PC, which using your own software based on our free LabVIEW driver can then be programmed to read the data at a few early channels (corresponding to the time before the trigger) and then ratio rest of the data curve to the average of these values (to correct for source fluctuations), with the resulting curve being averaged again. Given the above and data transfer times it is possible that only every other laser pulse would be used, but nonetheless it still represents a viable solution to your problem.