  We demonstrate long-lived coherence in internal hyperfine states of a single
\Ca{43} trapped-ion qubit $[T_2=1.2(2)\s]$, and in external motional states of
a single \Ca{40} trapped-ion qubit $[T_2'=0.18(4)\s]$, in the same apparatus.
The motional decoherence rate is consistent with the heating rate, which was
measured to be 3(1) quanta/sec. Long coherence times in the external motional
states are essential for performing high-fidelity quantum logic gates between
trapped-ion qubits. The internal-state $T_2$ time that we observe in \Ca{43},
which has not previously been used as a trapped-ion qubit, is about one
thousand times longer than that of physical qubits based on \Ca{40} ions. Using
a single spin-echo pulse to ``re-phase'' the internal state, we can detect no
decoherence after 1\s, implying an effective coherence time $T_2^{\mbox{\tiny
SE}} \gtish 45\s$. This compares with timescales in this trap for single-qubit
operations of \ish 1\us, and for two-qubit operations of \ish 10\us.
