How can I tell which band (2.4GHz or 5GHz) my iPhone's WiFi connection is on?

I have two access points for the same network in my house, with some overlap. One runs on 2.4GHz only, and one transmits both 2.4GHz and 5GHz.

I have an iPhone 5 (running iOS 6.0.1), which supports the 5GHz band and I'm wondering if there is a way for me to determine which frequency band it is connected to.

You can not tell from the iPhone directly (as of iOS version 7 native). You should be able to tell via the wireless router your connected to in it's logging functions, WiFi client list or some similar mechanism within the router. I can't provide you an exact process as every router is different. This answer assumes you do not want to JailBreak your iPhone. If you are not adverse to doing so there may be an app in the Cydia app store that will show more meaningful information with regard to WiFi frequency information.

You can however detect what cell tower frequency your phone uses by using Field Test mode. This will special mode allows users to find out a range of information about the phone’s connection to the network. In particular, the information displayed in the Field Test mode allows users to discover exactly which tower their phone is connecting to and which frequency band (WiFi) the signals are being sent on.

To access this mode:

• Access the iPhone's Field Test mode by typing *3001#12345#* in the phone’s keypad and press dial. This will bring up the Field Test menu.
• Select the “Cell Information” menu option to display a summary of all nearby towers. The “RX” code displays the signal strength of each tower. These RX values are dBm signal strength readings, where lower values means strong signal (so -60 signifies stronger signal than -80). The towers are listed by signal strength, and the top tower is usually the tower your phone is connecting to.
• Selecting the first tower will take you to a further menu that displays more specific information. The “Freq” value shows the frequency block that the network is using. If the Freq number is between 128 and 251, the tower is using the 850 MHz Cellular band. If the Freq number is between 512 and 810, the tower is using the 1900MHz PCS band.

Get an app on the App Store which will tell you the MAC address of the AP you're connected to. There are many, and they are free.

Then, enable the WiFi scanner in the AirPort Utility app: go to Settings > AirPort Utility to enable it. The scanner will tell you the MAC addresses of the detected APs and the frequencies used.

Since APs use different MACs for different frequencies, you can use the data provided to understand which frequency your device is using. It's complicated, but it is the only way to know, unless Apple opens its private APIs.

If your network is based on any Apple AirPort-thingy, you can set up a separate SSID for the 5GhZ-network and connect to that one manually on your iPhone 5, and from there be able to see which network you're on. To do this, follow the below step:

Airport Utility 6.0: Airport Extreme > Edit > Wireless > Wireless Options > Tick the box next to "5GHz network name".
Airport Utility 5.6 and earlier: Manual Setup > Wireless > Wireless Network Options > Tick the box. ^Source

P.S. Having a different SSID does not make it a separate network, it just makes the two bands differentiable.