The successor to Hubble, due for launch in 2018, is going to be put inside the giant thermal vacuum chamber where they tested the Apollo spaceships.
For 90 days, JWST will get a thorough check-up in the same airless, frigid conditions that will have to be endured when it eventually gets into orbit.
The telescope's aim will be to find the first stars to shine in the Universe.
To achieve this goal, the observatory is being given an enormous mirror and instruments that are tuned to detect some of faintest objects on the sky.
More than two decades in development, JWST is currently passing through the most critical phases on its construction timeline.
Any major technical hiccups now would seriously derail its launch readiness.
The past year has seen the assembly of the main telescope structure, with its 18 beryllium-gold mirror segments, at the US space agency's (Nasa) Goddard centre in Greenbelt, Maryland.
Attached to this structure are Webb's four observing instruments, which sit inside a cage on the rear of the big reflector.
The whole edifice has just gone through vibration and acoustic testing at Goddard. This has simulated the shaking and the roar of the carrier rocket. This will be an Ariane 5 provided by project partner, the European Space Agency.
One vibration run had to be stopped early when unexpected accelerations were detected in the region of the telescope's secondary mirror, but the analysis concluded it was not an issue of significant concern. So far, so good.
Now, JWST must fly to Nasa's Johnson centre at the end of April, or perhaps the beginning of May.
There, it will enter the famous Chamber A. This is the largest testing facility of its kind in the world, and the only one capable of housing an orbiting telescope on JWST's scale.
Engineers want to know that Webb's mirrors and instruments will function in unison when they are sitting out in space.
The chamber will provide important evidence of that by going super-cold (it can go as low as minus 260 Celsius).
An artificial light source shone on to Webb will ensure everything lines up as it should.
"When we are at temperature, we will unfold the mirrors and move them," explained JWST instrument systems engineer Bego?a Vila.
"On orbit, not only do they have to unfold, you have also to confirm the focus for the different instruments.
"If you think about it, with those 18 segments we start with 18 different images of a star, and so we have to move all those images until we have a single one and it is in focus. It's that process that we have to practise," she told BBC News.