Two labels in one caption possible?

Question

In a lot of journals it is common to have single digit figure numbers and denote parts in the figures as (a), (b), and so on. I wanted to use a similar style. After reading the documentation to the caption package and the cleveref package I did not find anything similar. I do not want to break up the picture in two parts and use subfigure with two captions but keep it all in one.

Is there an easy way to realize this? My example looks as follows without the correct references:

\documentclass{scrbook}

\usepackage{caption}
\usepackage{cleveref}

\begin{document}

\begin{figure}
  \centering
  a \rule{10pt}{10pt} \hspace{1cm} b \rule{15pt}{15pt} 
  \caption{Common text here. \label{part_a}\textbf{(a)} Description of a. \label{part_b}\textbf{(b)} Description of b. } 
\end{figure} 

This should read fig. 0.1a and fig. 0.1b but it is only \cref{part_a} and \cref{part_b}.

\end{document}

This gives this output:

Answer 1

A small addition to Mico's answer with \phantomcaption option:

\documentclass{article}
\usepackage{mwe}            %For dummy images
\usepackage{subcaption}
\usepackage{cleveref}
\crefdefaultlabelformat{\thesection.#2#1#3} 

\begin{document}
\begin{figure}
\centering
\begin{subfigure}{0.4\textwidth}
  \centering
  \includegraphics[width=0.3\textwidth]{example-image-a}
  \phantomcaption 
  \label{part_a}
\end{subfigure}
\hspace{1cm} 
\begin{subfigure}{0.4\textwidth}
  \centering
  \includegraphics[width=0.3\textwidth]{example-image-b}
  \phantomcaption
  \label{part_b}
\end{subfigure}
\caption{Common caption here \textbf{(a)} On left \textbf{(b)} On right} \label{fig-overall}
\end{figure} 

This should read fig.\ 0.1a and fig.\ 0.1b, and indeed it now reads \cref{part_a} and \cref{part_b}.
\end{document}

Answer 2

A good way to proceed, I believe, is to use the subcaption package and its subfigure environment. That way, you can assign labels to each subfigure (as well as, of course, to the entire figure).

Note that you need to assign a width to each subfigure environment. This could either be an absolute length, such as 3cm, or a fraction of a document variable such as \textwidth, as I've chosen to do in the MWE below. If you two subfigures and want to maximize their possible size, you could give them both a width of ca 0.48\textwidth, and use the command \hspace{\fill} to spread them apart. (You shouldn't use 0.5\textwidth as there's some "overhead" whitespace.)

\documentclass{scrbook}
\usepackage{caption,subcaption}
\usepackage{cleveref}

\begin{document}
\begin{figure}
\centering
\begin{subfigure}{0.3\textwidth}
  \centering
  a \rule{10pt}{10pt} 
  \caption{Description of a.} \label{part_a}
\end{subfigure}
\hspace{1cm} 
\begin{subfigure}{0.4\textwidth}
  \centering
  b \rule{15pt}{15pt} 
  \caption{Description of b. } \label{part_b}
\end{subfigure}
\caption{Common caption here} \label{fig-overall}
\end{figure} 

This should read fig.\ 0.1a and fig.\ 0.1b, and indeed it now reads 
\cref{part_a} and \cref{part_b}.

\end{document}