The original is Alfred Tarski's book "The completeness of elementary algebra and geometry", which was due to appear in 1940 but never made it into print because of the outbreak of WW2. An edition appeared after all in 1967 (Institut Blaise Pascal, Paris), but is not easy to come by.
Essentially the same argument is presented in Tarski's 1948/51 "A decision method for elementary algebra and geometry", available here.
Tarski uses an axiomatic setup of elementary geometry different (but equivalent) to Hilbert's, using only points (not lines or planes) as primitive terms, and two relation symbols, $B$ and $D$ (ternary and quaternary, respectively). $Bxyz$ signifies that the point $y$ is on the "line" $xz$ between the points $x$ and $z$ ("betweenness"), while $Dxyzw$ means that the distance between $x$ and $y$ equals that between $z$ and $w$ (the "equidistance" relation). Here, $x$, $y$, $z$ (and $w$) are allowed to coincide.
Full continuity is the second order axiom $\forall_{X}\forall_{Y}((\exists_{a}\forall_{x\in X}\forall_{y\in Y}Baxy)\rightarrow(\exists_{b}\forall_{x\in X}\forall_{y\in Y}Bxby))$, where $X$ and $Y$ are variables ranging over sets of points.
First order continuity is weaker, and is expressed as an axiom schema where $X$ and $Y$ are given as $\lbrace x\mid \phi(x)\rbrace$ and $\lbrace y\mid \psi(y)\rbrace$, respectively, for arbitrary first order formulas $\phi(x)=\phi(x,p_{1},\cdots,p_{n})$ and $\psi(y)=\psi(y,p_{1},\cdots,p_{n})$ that are allowed to contain parameters $p_{1},\cdots,p_{n}$.
A special case of first order continuity is the Circle Axiom, by which a line that contains an interior point of a circle (in the same plane) must meet that circle.
Completeness is the statement that any model $\mathfrak A$ of the axioms of elementary $n$-dimensional geometry without continuity is isomorphic to $K^{n}$ (with the obvious interpretations for the $B$ and $D$ relations) for a Pythagorean ordered field $K$ (that is $K\models\forall_{a}\forall_{b}\exists_{c}(a^{2}+b^{2}=c^{2})$), uniquely determined by $\mathfrak A$ up to isomorphism.
Under full continuity, $K$ must be $\mathbb{R}$, under first order continuity $K$ must be real closed, and for the Circle Axiom $K$ merely needs to be Euclidean (i.e., $K\models\forall_{a}\exists_{b}(a=b^{2}\vee -a=b^{2})$).
This is the content of the Representation Theorem, Th. I, (16.15) in "Metamathematische Methoden in der Geometrie" by W. Schwabhäuser, W. Szmielew and A. Tarski, Springer Hochschultext, 1983, an excellent reference for the metamathematics of elementary geometry (in German).
Edit: Let me add a few comments.
- The statement above that Tarki's setup is equivalent to Hilbert's is rather imprecise, as noted by Matt F. and others. Tarski works in first order logic, while a formalization of Hilbert's system is at least unclear. (Still, the axioms in Hilbert's axiom groups I-IV can be derived from Tarski's axioms, as shown in the Schwabhäuser, Szmielew, Tarski text).
- For the same reason, it is not clear what it would mean for Hilbert's system to be complete (in the modern sense), and I do not claim that "completeness" of Hilbert's system follows from that of Tarski. Hilbert includes a "completeness axiom", to the effect that his "model" of the axioms in groups I-V (where V is archimedeanity) cannot be extended to a "model" with a larger universe.
- To add to the confusion, my use of the word Completeness above (in the body of the answer, in reference to the Representation Theorem) was also unfortunate. Tarski has shown that the first order theory of real closed fields is complete (in the modern sense), and that, as a result, the same goes for the theory of $n$-dimensional elementary geometry (based on Tarski's axioms, with the first order continuity axiom schema included).
