The group, led by Jillian MacLean, has taken a 6,500sq ft site at the office building, which is located in Bishopsgate in the City of London.
Work will begin next month with the venue, which is to be called The Drift, due to open at the end of May.
Heron Tower, which will be 46 storeys high, will also feature a restaurant and sky bar on levels 38 to 40 when it opens later this year.
Shelley Sandzer, the property specialist, brokered the deal.
The group operates four sites, all in London. It opened the Folly in the City of London in November. It also operates the Anthologist, also in the City, the Parlour at Canary Wharf and the Refinery at Bankside.
Drake & Morgan is chaired by serial entrepreneur John Connell and backed by Imbiba Partnership, Connell's investment vehicle.
MacLean had worked at a number of sector companies such as Novus Leisure and Spirit Group, before launching the business three years ago.
Mark Stretton is editor of BigHospitality's sister publication M&C Report.