a constructor as a delegate - is it possible in C#?

I think as concise as you're going to get (without moving to a factory pattern) would be something with anonymous methods, like this:

delegate Foo FooGenerator(int x);

...    

void DoStuff()
{
    YourDelegateConsumer(x => new Foo(x));
}

This isn't doing strictly what you asked for (since you're passing a delegate to an anonymous method that returns a new instance, rather than a direct delegate to the constructor), but I don't think what you're asking for is strictly possible.

This is, of course, assuming you're using 3.5+

Nope, the CLR does not allow binding delegates to ConstructorInfo.

You can however just create your own:

static T Make<T>(Action<T> init) where T : new()
{
  var t = new T();
  init(t);
  return t;
}

Usage

var t = Make<Foo>( x => { x.Bar = "bar"; x.Baz = 1; });

I'm assuming you would normally do something like this as part of a factory implementation, where the actual types aren't known at compile-time...

First, note that an easier approach may be a post-create init step, then you can use generics:

static T Create<T>({args}) where T : class, ISomeInitInterface, new() {
    T t = new T();
    t.Init(args);
    return t;
}

You can then use MakeGenericMethod and/or CreateDelegate.

---

Otherwise; you can do this with on the fly with Expression (3.5) or DynamicMethod (2.0).

The Expression approach is easier to code:

    var param = Expression.Parameter(typeof(int), "val");
    var ctor = typeof(Foo).GetConstructor(new[] { typeof(int) });
    var lambda = Expression.Lambda<Func<int, Foo>>(
        Expression.New(ctor, param), param);
    var func = lambda.Compile();
    Foo foo = func(123);
    string s = foo.ToString(); // proof

or (using DynamicMethod):

    ConstructorInfo ctor = typeof(Foo).GetConstructor(new[] { typeof(int) });
    DynamicMethod dm = new DynamicMethod("Create", typeof(Foo),
            new Type[] { typeof(int) }, typeof(Foo), true);
    ILGenerator il = dm.GetILGenerator();
    il.Emit(OpCodes.Ldarg_0);
    il.Emit(OpCodes.Newobj, ctor);
    il.Emit(OpCodes.Ret);
    Converter<int, Foo> func = (Converter<int, Foo>)
        dm.CreateDelegate(typeof(Converter<int, Foo>));        
    Foo foo = func(123);
    string s = foo.ToString(); // proof