The Arcminute Microkelvin Imager (AMI) telescopes located at the Mullard Radio Astronomy Observatory near Cambridge have been significantly enhanced by the implementation of a new digital correlator with 1.2 MHz spectral resolution. This system has replaced a 750-MHz resolution analogue lag-based correlator, and was designed to mitigate the effects of radio frequency interference, particularly from geostationary satellites that contaminate observations at low declinations. The upgraded instrument consists of 18 ROACH2 Field Programmable Gate Array platforms used to implement a pair of real-time FX correlators -- one for each of AMI's two arrays. The new system separates the down-converted RF baseband signal from each AMI receiver into two 2.3 GHz-wide sub-bands which are each digitized at 5-Gsps with 8 bits of precision. These digital data streams are filtered into 2048 frequency channels and cross-correlated using FPGA hardware, with a commercial 10 Gb Ethernet switch providing high-speed data interconnect. Images formed using data from the new digital correlator show over an order of magnitude improvement in dynamic range over the previous system. The ability to observe at low declinations has also been significantly improved.