OK, here's a circuit that should work.
I have no idea what the amplitude of your noise source is. You might need more or less gain. You can still add gain in your first stage, since bandwidth should not be an issue. You could also add it before the filter, but be aware that the signal there is centered around the Veb breakdown voltage of the noise transistor, so adding gain after down sampling recreates the low frequency cutoff issue. I avoided that in the first amplifier (note C2, the 100nF cap). I was able to do this because the down sampling takes the higher frequency noise which is passed by U1a and folds it down below the sample rate. After the Sample and hold, any gain added has to have a very low frequency cutoff, probably in the milliHertz range, depending on your requirements.
In any case, the filter output as designed also has a big DC component, so you will probably have to cap-couple the output to get rid of the DC. When you do, be aware of the low frequency cutoff which cap coupling causes.
It might be better to use dual supplies, and remove the DC component before the gain stage. Then all the low frequency cutoff issues are moot.
What is your application?
Do you have the capability of using dual supplies?