Is C++ built on top of C?

A few C++ compilers (the original cfront, Comeau C++) use C as an intermediate language during compilation. Most C++ compilers use other intermediate langauges (e.g. llvm).

Edit: Since there seems to be some misunderstanding about the history: "C with classes" started out using a preprocessor called "Cpre". At that time, it was seen strictly as a dialect of C, not a separate language in itself. In December 1983, people were starting to view it as a separate language, and the name C++ was invented. As it happens, development of cfront started in April 1983, so a reasonably usable version became available (to a select few) just about the same time as the name "C++" came into use. This appears to be mostly coincidence though.

As far as producing C as its output, that was really quite common on Unix. Just for example, the Berkeley Pascal compiler and at least a couple of Fortran compilers also produced C as their output.

There is, however, a huge difference between Cpre and Cfront. Although both produced C as their output, Cpre did virtually no syntax checking of its own -- it looked for a few specific things, and did a relatively mechanical translation on them. It wasn't until the C compiler looked at the result that real syntactical analysis was done. If your code contained a syntax error, it was almost certain that it wouldn't be caught until the C compiler parsed the output from Cpre.

Cfront, however, did full syntactical analysis of the source code itself, so (short of a bug in its code generator) you'd never see a syntax error from the C compiler. The C compiler was simply used as a code generator so nobody needed to rewrite CFront to accommodate different processors, object file formats, etc.

If you want to get into more detail, chapter 2 of The Design and Evolution of C++ is devoted almost entirely to the "C with Classes" time frame (and there are various other details about it spread throughout the book).

In the early days of C++ compilers, some did it that way. I haven't seen a C++ compiler implemented that way since the late 1980s however.

Not in most modern compilers.

The original C++ compiler was actually a preprocessor however. It generated C code, which was then compiled by a C compiler.

No, but like most myths there's a shred of truth to this. The original compiler for C with classes (which later became C++) was nicknamed CFront and did translate to C.