Due to the discussion around this question, I've done some more digging to find out whether or not the epoxy layer shown in pericynthion's answer can be considered a structural part of the tank.
In many NASA documents, the epoxy layer isn't mentioned:
NASA applied two basic types of Thermal Protection System materials to the ET. One type was a low-density, rigid, closed-cell foam. This foam was sprayed on the majority of the tank’s “acreage”—larger areas such as the liquid hydrogen and liquid oxygen tanks as well as the intertank—also referred to as the tank “sidewalls.”
The other major component was a composite ablator material (a heat shield material designed to burn away) made of silicone resins and cork.
NASA Facts - External Tank Thermal Protection System doesn't mention it either:
There are two basic Thermal Protection Systems on the External Tank: One is low-density, closed-cell foam; the other is a denser composite material called ablator, made of silicone resins and cork. An ablator is a material that dissipates heat by eroding.
The closed-cell foam used on the tank acreage is a Spray-On- Foam-Insulation often referred to by its acronym as SOFI (pro- nounced sō-FEE). The composite material is Super Lightweight Ablator, known as SLA (pronounced slaw).
The External Tank uses ablators on areas that are subjected to extreme heat, such as the aft dome near the engine exhaust and on protuberances that are exposed to aerodynamic heating, such as the cable trays.
I have found references that mention an epoxy primer.
The application of the (epoxy) primer, used largely as an adhesive for foam insulation,...
It seems unlikely to me that you'd need a layer of primer that's 12 mm thick, but I'm no expert in this area.
The Thermal Protection Systems document contains this graphic:
which lists a resin/cork combination as the ablator, used in high heating areas. From the graphic this ablator appears to be used instead of SOFI foam, not underneath it.
The resin/cork material was applied in sheets. This material caused some problems before the first launch, with panels coming off the tank as it contracted when it was filled with cryogenics.
Provisionally, I'd have to conclude that the epoxy layer is not a structural part of the tank.