Having a master page associated with an ASPX page does not prevent it from being placed in the _layouts directory. All of my ASPX pages use a custom master page I wrote, and they're all stored in the _layouts directory. But there's enough methods to hide the breadcrumbs without changing the storage location.
If you want to hide the breadcrumbs on only a specific page, the simplest method to do so is to prevent the contents of the breadcrumbs from being displayed. On the default master page (and I'm going assume that your master page also has this), there is a content called "PlaceHolderTitleBreadcrumb", which is what displays the breadcrumb. By overriding this in the ASPX file, you can prevent this from displaying. A simple way is to just insert the following line in your ASPX file:
<asp:Content ContentPlaceHolderId="PlaceHolderTitleBreadcrumb" runat="server"> </asp:Content>
By being in the page itself, it becomes a custom version of the breadcrumb instead of the master page's, so it won't evaluate itself and will leave you with a blank line. The will keep content inside the table cell, so it will specifically make a blank line occupy the same space that the breadcrumbs normally would. If you just put an empty content there, then it will instead render nothing there and pull all of your content upwards. Up to you if you think that looks better.
As far as where to place the line, I recommend just beneath the following lines, if your ASPX page has them. If you don't have the following lines, you might have to experiment a bit to find what position works properly.
<asp:Content ContentPlaceHolderId="PlaceHolderPageImage" runat="server">
<IMG SRC="/_layouts/images/blank.gif" width=1 height=1 alt="">
</asp:Content>