Following the answers (thanks!) things now appear a little bit clearer to me.
My problem is the high amount of storage because the measurement time is about 2-3 minutes. That means the necessity of about 150s*1MB/s=150MB. That's why internal storage is far away from being great enough.
Therefore I need external RAM or SSD. Has anyone got experience with connecting SSD and DUE?
That is a pretty tough target for a Due. SSDs usually have a PCI or SATA interface, not available on Due. At a pinch you might be able t o use a USB mass storage device, not sure that would give a better performance than SD.
Serial SD is quite slow, the SAM3X has a high-speed SD interface, not used on Due. I have seen it in use on the Duet board though, a Due variant designed for 3D printing. SD cards tend to have long unpredictable pauses anyway because of their onboard controller.
Unfortunately the Due does not bring out the external memory bus on the SAM3X, which would allow easy access to large RAM or Flash.
I think your best bet for a DIY solution with the Due would be an SPI connected NOR Flash, e.g. http://uk.rs-online.com/web/p/flash-memory-chips/8096901/. You will need at least 3 to get 150MB, I would connect 4 and interleave writes so you should be able to achieve 1MB/sec. SPI bus will be a bottleneck I think.
Otherwise you could look at the Duet, it is mostly Arduino compatible. There is nothing else suitable in the Arduino environment I can think of. The boards with big memories (IOW, Linux capable) require some serious bare-metal coding. I know there is some bare-metal stuff for Raspberry Pi, that might be worth looking at.