  Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct
an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild
complex, which arises as a totalization of a double complex having Hochschild
chains as negative columns and Hochschild cochains as non-negative columns. We
prove that the complex $\mathcal{D}^*(A,A)$ computes the singular Hochschild
cohomology of $A$. We construct a cyclic (or Calabi-Yau) $A$-infinity algebra
structure, which extends the classical Hochschild cup and cap products, and an
$L$-infinity algebra structure extending the classical Gerstenhaber bracket, on
$\mathcal{D}^*(A,A)$. Moreover, we prove that the cohomology algebra
$H^*(\mathcal{D}^*(A,A))$ is a Batalin-Vilkovisky (BV) algebra with BV operator
extending Connes' boundary operator. Finally, we show that if two Frobenius
algebras are quasi-isomorphic as dg algebras then their Tate-Hochschild
cohomologies are isomorphic and we use this invariance result to relate the
Tate-Hochschild complex to string topology.
