  H-atoms in H$_2$O are rearranged by strong optical fields generated by
intense, 10 fs laser pulses to form H$_2^+$, against prevailing wisdom that
strong fields inevitably lead to multiple molecular ionization and the
subsequent Coulomb explosion into fragments. This atomic rearrangement is shown
to occur within a single 10 fs pulse. Comparison with results obtained with
$\sim$300-attosecond long strong fields generated using fast Si$^{8+}$ ions
helps establish thresholds for field strength and time required for such
rearrangements. Quantum-chemical calculations reveal that H$_2^+$ originates in
the $^1$A state of H$_2$O$^{2+}$ when the O-H bond elongates to 1.15 a.u. and
the H-O-H angle becomes 120$^o$. Bond formation on the ultrafast timescale of
molecular vibrations (10 fs for H$_2^+$) has hitherto not been reported.
