Is this for a home project or commercial application? What exactly are you trying to do? Reason for asking is any instantenous change on the AC mains will not show up on the DC output. You have a 220 uF cap in there that will not follow change very quickly. Additionally you will have AC ripple. The greater the cap value the less ripple but the slower to respond to AC mains changes. While the circuit will work it is not the best way to go about things. The cap will also charge to E peak so if you want the RMS value you will I assume calibrate for RMS with your pot? How fast you sample is just a matter of how fast you want to update the data as you are dealing with a DC level that is in theory proportional to an AC signal level. All of this also assumes a true sine wave being measured.
There are much better ways to condition the signal, some costly and some not as costly depending on exactly what you want and what the goal is.
Not a useful circuit because two forward bias diode drops subtract from 1.4*9V, making the resulting DC output highly non-linear with respect to the peak AC input voltage.
I would use the 9V transformer, but run it through an opamp precision rectifier. Provides rapid peak acquisition, accuracy, and controllable decay.
Remove the capacitor.No need any capacitors.So now you get unsmoothed low level DC voltage.
Make a zero cross detection circuit (you can do with a single transistor).Detect every ZX & turn ON a timer for 5mS.Every middle in the semi period you will get the peak value.Read AD at this time & compare the results.
Use an opamp precision rectifier as Mike suggested. If you desire DC you can run that signal through an RC filter, which will give the average value (better than measuring the peak since it's less affected by waveform distortion and line spikes).
If you want true RMS then you will need to sample the waveform multiple times over each cycle and perform the RMS calculation with the micro (square each sample, sum the samples, and then take the square root of the sum).