Steven Smale labels the following statement "The Basic Theorem of Morse Theory" in A Survey of some Recent Developments in Differential Topology:

Let f be a $C^\infty$ function on a closed manifold with no critical points on $f^{-1}[-\epsilon,\epsilon]$ except k nondegenerate ones on $f^{-1}(0)$, all of index $s$. Then $f^{-1}[-\infty,\epsilon]$ is diffeomorphic to $X(f^{-1}[-\infty,-\epsilon];f_1,\ldots,f_k;s)$ (for suitable f_{i}.

Here $X(M;f;s)$ for $f\colon\,(\partial D^s)\times D^{n-s}\to M$ is M with an s-handle attached by f.

Where can I find a complete proof of this theorem, with all the t's crossed and i's dotted? Textbooks (Milnor, Matsumoto) only seem to prove homology/homotopy versions of the above statement, usually with substantial steps to be filled in by the reader. I nosed around some old papers for a few hours, (surely Smale himself proved it somewhere!) but to no avail. If I were to continue to search, no doubt I could eventually turn it up (there are a finite number of differential topology papers written 1958-1962, which is when I assume it was proven), but because I think that this question might be of wider interest, and to save me a lot of time, I'd like to ask:

Where can I find a complete unabridged proof of "The Basic Theorem of Morse Theory"? (in fact I care only about low dimensions) What is the original paper, and is there a textbook exposition of it anywhere?