PottedMeat is on the right track. We don't have graphics, so here's how to do it with a single power supply, a zener diode and an op-amp:
1. Bias the zener diode by connecting it to a voltage source through a resistor as shown in his diagram. A zener voltage around 6 to 8 volts with a nominal zener current around 5 mA would be a good value.
2. Use an op-amp as a buffer to provide a low output impedance. Configure an op-amp as a voltage follower by connecting the output to the inverting (-) input. The op-amp must have sufficient bandwidth and be stable at unity gain. A good, inexpensive choice would be an LF351 (National) or TL071 (Texas Instruments).
3. Connect the non-inverting (+) input to the junction of the resistor and the zener.
4. Connect a 50 ohm resistor from the output of the op-amp to positive side of an output capacitor. Connect the other lead of the capacitor to a "tie down" resistor (around 100 kOhms) to ground. The value of the capacitor depends on the input impedance of the test equipment. If the input impedance is greater than 10 kOhms, a 22 uF capacitor will be sufficient. The junction of the capacitor and the tie down resistor is your output terminal.
5. Power the op-amp with either single positive or a bipolar voltage supply. +12 to 15 volts or +/- 12 to 15 volts would be a good value for the power supply. Be sure to add a decoupling cap (0.1 uF ceramic) from the power supply (pin 6) to ground. This will help prevent the op-amp from oscillating.
If you happen to have a dual op-amp (LF353 or TL072), you can safely defeat the second amp by configuring it as a voltage follower and grounding the non-inverting input.
If you need to amplify the white noise signal, you will need a few more components The easiest way to allow for this would be to power the op-amp with a bipolar supply and operate the circuit around 0 volts DC. To do this, add a coupling cap from the junction of the zener diode and bias resistor to the non-inverting input of the op-amp and a "tie down" resistor from the non-inverting input of the op-amp to ground. The cap can be in the range of 22 uF. The resistor can be between 10 kOhms and 100 kOhms.
This will center the AC (noise) signal around ground, so if you amplify the signal, you will not also amplify the DC reference voltage beyond the limits set by the power supply.