= Solution
Write $X=\operatorname{Pic}^{g-1}(C)$. The <theta divisor> is the effective locus
$$
\Theta=\{[M]\in X:H^0(C,M)\ne0\}=\operatorname{im}\bigl(C^{(g-1)}\longrightarrow X\bigr).
$$
For its <scheme> structure, take a fixed effective <Cartier divisor> $E$ of degree $e\ge g$ and a universal <line bundle> $M$ locally on $X$. Both $\pi_*M(E)$ and $\pi_*(M(E)|_E)$ are rank-$e$ <vector bundles>; the first assertion follows from <Serre duality> and the <Riemann-Roch theorem>. Define $\Theta$ by the <determinant> of their evaluation map. Its kernel is $H^0(C,M)$, so this is precisely the required locus. Exact sequences for enlarging $E$ cancel an invertible determinant factor, making the <determinant description of the theta divisor> independent of $E$.
This is a genuine irreducible reduced <Cartier divisor>. Choose $g-1$ points imposing independent conditions on $H^0(C,K_C)$. For their effective <Cartier divisor> $D'$, the <Riemann-Roch theorem> gives $h^0(\mathcal O(D'))=h^1(\mathcal O(D'))=1$. These points form a nonempty open set in the irreducible <symmetric product of a curve> $C^{(g-1)}$; the <Abel map of an algebraic curve> is generically one-to-one there, so its image has dimension $g-1$. At such a class $M$, the first-order change of the evaluation <determinant> is the cup-product functional
$$
H^1(C,\mathcal O_C)\longrightarrow\operatorname{Hom}(H^0(C,M),H^1(C,M)).
$$
Under <Serre duality> its dual is multiplication $H^0(M)\otimes H^0(K_C\otimes M^{-1})\to H^0(K_C)$. The product of two nonzero sections on an integral <algebraic curve> is nonzero. Thus the functional is nonzero and the <determinant> has multiplicity one generically. Its irreducible support is consequently the reduced <theta divisor>.
Now let $L=\mathcal O_C(D)$ have degree $g$. The <Riemann-Roch theorem> gives $h^0(L)\ge1$, while evaluation at $P$ gives
$$
H^0(C,L(-P))=\ker\bigl(H^0(C,L)\longrightarrow L|_P\bigr).
$$
If $h^0(L)\ge2$, this kernel is nonzero for every $P$, so the entire image of $\phi_D$ lies in $\Theta$. If $h^0(L)=1$, write $s$ for a nonzero section. The kernel is nonzero exactly at the zeros of $s$, so the image is not contained in $\Theta$. Moreover $h^1(L)=0$, and the family $L(-P)$ is computed by the rank-one evaluation map $H^0(L)\otimes\mathcal O_C\to L$. Its <determinant> is $s$ itself. This proves the scheme-theoretic <restriction of the theta divisor to an Abel curve>:
$$
\boxed{\phi_D(C)\subseteq\Theta\iff h^0(L)\ge2,\qquad \phi_D^*\Theta=\operatorname{div}(s)\text{ when }h^0(L)=1.}
$$
Here the intersection is regarded as a <Cartier divisor> on $C$. Since $D$ was specified as a <divisor class>, it means its unique effective representative, with all multiplicities. The map $\phi_D$ is an embedding: distinct points with the same image would give a degree-one map $C\to\mathbb P^1$; its cotangent map is evaluation of canonical differentials, which is surjective because $K_C$ is base-point-free. To see the latter, $h^0(\mathcal O_C(P))=1$ for genus at least one, and the <Riemann-Roch theorem> gives $h^0(K_C-P)=g-1$. Thus the pullback also describes the intersection on the embedded curve.
It remains to count sections on $X$. Let $s_\Theta$ be the section defining the <theta divisor>, and take any $s\in H^0(X,\mathcal O_X(\Theta))$. The quotient $r=s/s_\Theta$ is a <rational function>. Choose a fixed degree-$2g-1$ <line bundle> $T$ and the morphism
$$
\beta:C^g\longrightarrow X,\qquad (P_1,\ldots,P_g)\longmapsto[T(-P_1-\cdots-P_g)].
$$
It is surjective: for every $M\in X$, the degree-$g$ bundle $T\otimes M^{-1}$ has a nonzero section by the <Riemann-Roch theorem>. On a general coordinate curve, put $D=T-\sum_{j\ne i}P_j$. The $g-1$ general points impose independent conditions on the $g$-dimensional space $H^0(C,T)$, leaving $h^0(\mathcal O(D))=1$. This independence follows inductively by choosing each new point away from the common zero set of the remaining nonzero sections. The restriction already proved identifies $\mathcal O_X(\Theta)$ on that coordinate curve with $\mathcal O_C(D)$, whose section space is one-dimensional. Hence $r\circ\beta$ is constant along every general coordinate curve.
A <rational function> on a product of integral curves which is constant in each factor is constant: on a common dense open set, successively varying one coordinate identifies its value with the value at a fixed general tuple. Therefore $r\circ\beta\in k$, and surjectivity of $\beta$ gives $r\in k$. Every $s$ is a scalar multiple of $s_\Theta$, proving
$$
\boxed{\dim H^0(X,\mathcal O_X(\Theta))=1.}
$$
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