See sections 43, Analog-to-Digital Converter (ADC), and 44, Digital-to-Analog Converter Controller (DACC), in the Atmel-11057-32-bit-Cortex-M3-Microcontroller-SAM3X-SAM3A_Datasheet.pdf specsheet.
In particular, subsection 43.2, Embedded Characteristics, says: “1 MHz Conversion Rate”. Some time diagrams appear quite a few pages later that suggest 10-bit conversions can be done more rapidly than 12-bit.
For the DAC (or DACC, per the specsheet), section 44.6, Functional Description, says:
The DACC uses the master clock (MCK) divided by two to perform conversions. This clock is named DACC Clock.
Once a conversion starts the DACC takes 25 clock periods to provide the analog result on the selected analog
AIUI, the Due's master clock frequency is 84 MHz, so DACC Clock frequency is 42 MHz, whence 25 cycles take 575 ns, which is an update frequency of about 1.74 MHz.
For more details about the ADC, see eg
Analog performance, particularly of Due's 12 bit ADC at djerickson.com.