amplify small dc signal sitting on large dc offset

The sensor is an ML8511 UVA sensitive device, although this problem could relate to any number of situations?

The datasheet suggests that the offset is 1V to within +-50mV, so it might just be simplest to remove a fixed 1V offset and then only have a gain of 20, i.e. 2 mV +- 50 mV would fit in a 2V range for a high-res ADC. which could be (say) 24-bit since you mention the wanted signal is very static too. This might all be simpler than trying to sample-and-hold, although if there are periods where it's known that there is no input, (could even be a shutter, perhaps motor spun) then the sampling could be done frequently.

If up to 16-bit is sufficient then ADS1115 or MCP3426 etc is very low-cost and has a built-in amplifier.

Otherwise, I don't know how slowly-changing the signal is; is a filter ruled out?

One method is a bit like a data slicer, i.e. using just an op-amp with a RC filter on one input, and the op-amp set to a gain. 

Yes well I am just gathering ideas how to tackle the problem and the only tests I have carried out is to determine that very low levels of UVA can be detected/measured with the ML8511. 
The main issue is that even attempting to test the linearity at low UV levels was hampered by the 1V offset overloading the measuring equipment.
The choice of A/D is already decided as the code is written and operational using an arduino microprocessor and so I need to try to interface the ML8511 to this platform.
It may be that just using a low noise differential amplifier will be OK but I just thought I would 'throw it open' to the community in case someone had encountered a similar problem and come up with a novel solution :-)

Dave

Offset only affect the display. It doesn't improve accuracy or resolution.

I don't know how much money you have to throw at this task, but a method could be to turn it into AC, by using a mirror to bounce the UVC into the sensor. The mirror could be attached to a voice coil, deflecting it tens or hundreds of times a second (for instance).

A real mirror is several hundred $$$ but according to this chart (google images search) aluminium reflects very well, so a decent but still low-cost mirror might work reasonably well.

You could try either 1 or 2 sensors in a Wheatstone bridge configuration to minimize power supply and thermal fluctuations. Then a differential amp to amplify the low signal.

Building on Shabaz' idea, maybe you could modulate the light source (with a shutter or encoder wheel). This would allow you to compare the UV reading to the no UV reading, regardless of drift of the offset.

And then for the ultimate performance (maybe overkill, but I don't know what the end use-case is) implement a lock-in amplifier (synchronous demodulator) either in the Arduino (likely hard if it's an Uno, but feasible in more powerful Arduinos theoretically) or externally. It's not difficult, but not a weekend project. But at least these AC conversion ideas open up the possibility of it, if it were ever required.

What do you see on pin 10 if you measure it? That's where the internal reference comes out to the smoothing cap. If you're lucky, it's the same as the offset. If not, you could probably scale it outside the chip to match whatever's going on inside.

Well spotted! I was wondering where was the Vref out like any decent analog sensor should have. The datasheet mentions TR being Vref so it looks perfect.

It seems some breakout boards for it, like Sparkfun, didn't think of bringing that out to a pin, but it could easily be wired by soldering onto the capacitor if this is a breadboard prototype etc.