Using an External ADC with Arduino
Yes, if you have a 0 to 30 mV signal that you need to measure to an accuracy of 2 microvolts, you'll need an ADC with an effective number of bits (ENOB) of at least 13.9 bits and an amplifier to make the signal to match the range accepted by the ADC.
Most ADCs pretty much require some sort of op-amp or differential amplifier between such low-level input signals and the ADC input pins.
A few ADCs have a differential amplifier already built in -- that makes things a lot easier to get working, but I don't know if any of such ADC available would meet the other requirements of your project.

I don't know any way to make such high-resolution measurements other than using a dedicated external ADC chip.

You'll want to read the ADC datasheet -- some of them, and so far all the microcontrollers I've seen that claim to have a built-in "16 bit ADC", have an effective number of bits (ENOB) worse than the 13.9 ENOB you need at the sampling rates I typically use.

Adafruit has a very nice tutorial describing how to connect the 12-bit ADC ADS1015 or the 16-bit ADC ADS1115 to an Arduino.

The Arduino playground has a brief tutorial describing connecting a 12-bit ADC MCP3208 or a 13-bit ADC MCP3304 to an Arduino.

I see from the Arduino blog and other sites that people have connected some even higher resolution ADCs to an to an Arduino board, including
the 24-bit Linear LTC2440 ADC;
the 18-bit LTC2400 ADC (24 bit at 5 samples per second); and
the 24-bit ADC TI ADS1220.


Arduino blog. "Archive for the ‘ADC’ Category"
Peter Oakes. "Arduino UNO, Using precision ADCs and DACs". (video tutorial: connecting the 24-bit ADC TI ADS1220 to an Arduino).
Martin Nawrath. "Connect a LTC2400 High Precision 24 Bit Analog to Digital Converter ... to the Arduino"
John Boxall. "Tutorial – Arduino and the TI ADS1110 16-bit ADC".


Sounds like a fun project.
Good luck.

EDIT:


Steve Luce. The LTC2440 24 BIT ADC and the Arduino Uno.


Everyone that uses an ADC needs some antialiasing filter between the signal and the ADC.

People that measure very faint signals usually need some sort of amplifier between the signal and the ADC.
You may find the list of suggested op amps on p. 7 of the AMP03 datasheet useful.

Because historically ADCs were expensive (and high-resolution ADCs were unavailable), people have developed many clever tricks for processing signals before feeding the processed signal into a relatively low-resolution ADC.
(Lock-in amplifiers, chopper amplifiers, manual precision gain adjustment, manual precision offset adjustment, heterodyning, etc.)

Have you tried asking "What sort of amplifier do I need to measure ((something)) with a 16-bit ADC" at https://electronics.stackexchange.com/ to learn some of those tricks that specifically apply to ((something))?