You could try:
\begin{samepage}
This is the first paragraph. This is the first paragraph.
This is the first paragraph. This is the first paragraph.
\nopagebreak
This the second. This the second. This the second.
This the second. This the second. This the second.
This the second. This the second.
\end{samepage}
samepage prevents LaTeX from pagebreaking within one paragraph, i.e. within the samepage environment, pagebreaks are only between paragraphs. Thus, you need nopagebreak as well, to prevent LaTeX from pagebreaking between two paragraphs.