Determining DC gain of single supply op-amp (no AC component)

The first which comes to mind is bilateral switches like 74HC4066. They have a few tens of Ohms of on-resistance, but that won't probably much of a problem if your feedback resistor values are high enough, otherwise take them in account, and for example use a 950\$\Omega\$ resistor instead of a 1k.
I'm getting some rather negative reactions on this, but I just wanted to make the same remark endolith makes: if Matt can use a symmetrical power supply the switch can be placed on the side of the virtual ground, and especially when using a high feedback resistance there will be hardly any voltage drop over it, so also no distortion. If the feedback resistance is 10k\$\Omega\$ then the resistance of the switch is only 1/10th of the tolerance on a 1/4W E12 resistor.
FakeName suggests some AD devices as better alternatives to the 74HC4066. Particularly the ADG1401 he mentions looks very interesting, with an \$R_{ON}\$ < 2\$\Omega\$. (Thanks, Fake)
Matt already seems to have found Maxim analog multiplexers, but I want to include the MAX4781, which has similar \$R_{ON}\$, in case others are interested. The Intersil ISL84781 is a pin-for-pin replacement for the MAX4781, and has a maximum \$R_{ON}\$ of 0.8\$\Omega\$.

High end audio equipment sometimes uses reed-relays here, which have the advantages of a mechanical switch: m\$\Omega\$ resistance when on, G\$\Omega\$ when off.

edit2
Just had another idea! :-) An OTA (transconductance amplifier) can be used to make a VCA (Voltage Controlled Amplifier). This one uses an LM13700:

Instead of having a series of digital control lines controlling switches you vary the amplification with an analog voltage. DAC can be as simple as R-2R ladder network.

edit3
Yet another solution also takes the problem one level up, like the OTA: you don't really want switches, you want a variable gain amplifier, and therefore the feedback resistor must be variable. Like with a digital potentiometer. This one has a 100k\$\Omega\$ range in 400\$\Omega\$ steps. You can also choose a smaller range in combination with a fixed resistor, of course. Possible drawback here is that the wiper has a rather high resistance of 800\$\Omega\$ (maximum), though this just means you'll always have the two lowest steps.

If the signal out of the dual-supply op-amps is also bipolar around the virtual ground rail, the single-supply parts will face challenges with such signal - unless clipping at the virtual ground rail is the design intent.

Assuming that all signal in the single-supply op amp portion of the design is definitely above the virtual ground potential, the "new stage" does not use some internal-to-the-chip negative rail, it uses the virtual ground as its most negative supply rail.

One point of consideration for such designs is to ensure that the virtual ground itself is a very low-impedance rail with sufficient current sourcing and sinking capacity. Use an op-amp based rail splitter with hefty decoupling capacitors between all rails, and perhaps add a BJT stage, to achieve this.

Alternatively, use a purpose-built rail splitter part such as the Texas Instruments TLE2426 for the virtual ground.

Also note that your single-supply stage output will be bound between Virtual Ground and the positive rail, not utilizing the lower half of the possible voltage range. In other words, output signal peak to peak is limited to ~4.5 Volts (assuming RRO OpAmps) and not ~9 Volts.