The basic dynamic of the Singularity is positive feedback - smarter minds are better at inventing even smarter minds. Intelligence creates technology which enhances intelligence. While the process of self-enhancement must eventually reach a limit for any given piece of hardware, this bottleneck only exists until the transhuman AI can assist in the development of new hardware. Or produce any other one-kilogram object, from diamond to hamburger, given a complete specification and a feedstock containing the needed atoms. This is more than enough to run an AI, even a transhuman AI, at a millionfold or greater subjective speedup.
Nanotechnology is a field that has matured considerably in the past decade.
Companies such as Zyvex are pioneering the automation of what tools exist. As the specialized capabilities improve, the remaining barriers to true molecular manufacturing become conceptual rather than material - the pieces exist, but not the knowledge of how to put them together. In a few years, the time required for a transhuman AI to create the first nanodevices might be measured in weeks or days rather than months or years, after which the AI could move to nanocomputing hardware and upgrade itself to full-scale superintelligence. The diamondoid making up a nanodevice, held together by molecular bonds, is considerably stronger than biological proteins held together by the electrostatic forces that operate between amino acids. With that degree of control over the lower levels of biology, almost arbitrary results can be produced from a high-level viewpoint.