To simplify the expression

\[
S = \frac{2}{1 \times 2} + \frac{2}{2 \times 3} + \frac{2}{3 \times 4} + \ldots + \frac{2}{18 \times 19} + \frac{2}{19 \times 20},
\]

we start by noticing that the general term can be expressed as:

\[
\frac{2}{n(n+1)}.
\]

This can be simplified using partial fractions:

\[
\frac{2}{n(n+1)} = \frac{2}{n} - \frac{2}{n+1}.
\]

Thus, we can re-write the entire sum \(S\):

\[
S = \left( \frac{2}{1} - \frac{2}{2} \right) + \left( \frac{2}{2} - \frac{2}{3} \right) + \left( \frac{2}{3} - \frac{2}{4} \right) + \ldots + \left( \frac{2}{19} - \frac{2}{20} \right).
\]

When we add these, we observe that all the intermediate terms cancel out. The only terms remaining are the first from the first fraction and the last from the last fraction:

\[
S = 2 - \frac{2}{20} = 2 - 0.1 = 1.9.
\]

Thus, the value of the sum \(S\) is:

\[
\boxed{1.9}.
\]