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There are many parsing rules that cannot be parsed by a bison/yacc parser (for example, distinguishing between a declaration and a function call in some circumstances). Additionally sometimes the interpretation of tokens requires input from the parser, particularly in C++0x. The handling of the character sequence >> for example is crucially dependent on parsing context.
Those two tools are very poor choices for parsing C++ and you would have to put in a lot of special cases that escaped the basic framework those tools rely on in order to correctly parse C++. It would take you a long time, and even then your parser would likely have weird bugs.
yacc and bison are LALR(1) parser generators, which are not sophisticated enough to handle C++ effectively. As other people have pointed out, most C++ compilers now use a recursive descent parser, and several other answers have pointed at good solutions for writing your own.
C++ templates are no good for handling strings, even constant ones (though this may be fixed in C++0x, I haven't researched carefully), but if they were, you could pretty easily write a recursive descent parser in the C++ template language. I find that rather amusing.
It will probably take you years, and you'll probably switch to some other parser generator in the process.
Parsing C++ is notoriously error-prone. The grammar is not fully LR-parsable, as many parts are context-sensitive. You won't be able to get it working right in flex/yacc, or at least it'll be really awkward to implement. There are only two front-ends I know of that get it right. Your best bet is to use one of these and focus on writing the back-end. That's where the interesting stuff is anyway :-).
Existing C++ Front Ends:
The EDG front-end is used by most of the commercial vendors (Intel, Portland Group, etc.) in their compilers. It costs money, but it's very thorough. People pay big bucks for it because they don't want to deal with the pain of writing their own C++ parser.
GCC's C++ front-end is thorough enough for production code, but you'd have to figure out how to integrate this into your project. I believe it's fairly involved to separate it from GCC. This would also be GPL, but I'm not sure whether that's a problem for you. You can use the GCC front-end in your project via gcc_xml, but this will only give you XML for classes, functions, namespaces, and typedefs. It won't give you a syntax tree for the code.
Another possibility is to use clang, but their C++ support is currently spotty. It'll be nice to see them get all the bugs out, but if you look at their answered 2009-12-25T19:08:44.133
A few years - if you can get research grant to re-write new lex/yacc :-)
People keep chasing their tails on this a lot - starting with Stroustrup who was always fancied being a language "designer" rather than actual compiler writer (remember that his C++ was a mere codegen for ages andwould still be there if it wasn't for gcc and other folks).
The core issue is that real research on parser generators pretty much ceased to exist ever since CPU-s became fast enough to handle functional languages and brute-force recursive descent. Recursive descent is the last resort when you don't know what to do - it does exhaustive search till it nabs one "rule" that fires. Once you are content with that you kind of loose interest in researching how to do it efficiently.
What you'd essentially need is a reasonable middle-ground - like LALR(2) with fixed, limited backtraching (plus static checker to yell if "desiogner" splurges into a nondeterministic tree) and also limited and partitioned symbol table feedback (modern parser need to be concurrency-friendly).
Sounds like a research grant proposal, doesn't it :-) Now if we'd find someone to actually fund it, that would be something :-))