We show the complete integrability of the string propagation in $D$-dimensional de Sitter spacetime. We find that the string equations of motion, which correspond to a noncompact $\mathrm{O}(D, 1)$-symmetric $\ensuremath{\sigma}$ model, plus the string constraints, are equivalent to a generalized sinh-Gordon equation. In $D=2$ this is the Liouville equation, in $D=3$ this is the standard sinh-Gordon equation, and in $D=4$ this equation is related to the ${B}_{2}$ Toda model. We show that the presence of instability is a general exact feature of strings in de Sitter space, as a direct consequence of the strong instability of the generalized sinh-Gordon Hamiltonian (which is unbounded from below), irrespective of any approximative scheme. We exhibit B\"acklund transformations for this generalized sinh-Gordon equation, which relate expanding and shrinking string solutions. We find all classical solutions in $D=2$ and physically analyze them. In $D=3$ and $D=4$, we find the asymptotic behaviors of the solutions in the instability regime. The exact solutions exhibit asymptotically all the characteristic features of string instability: namely, the logarithmic dependence of the cosmic time $u$ on the world sheet time $\ensuremath{\tau}$ for $u\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$, the stretching (or the shrinking) of the proper string size, and the proportionality between $\ensuremath{\tau}$ and the conformal time.