MVC 5 Multiple Models in a Single View

I would say this is good example of using ViewModel here. I would suggest something like –

Create ViewModel with the composition of the two classes

public class AddWeightModel
{
    [Required]
    [DataType(DataType.Text)]
    [Display(Name = "Stone")]
    public Nullable<short> Stone { get; set; }

    [Required]
    [DataType(DataType.Text)]
    [Display(Name = "Pound")]
    public Nullable<short> Pound { get; set; }
}
....
public partial class Weight
{
    public int Id { get; set; }
    public string UserId { get; set; }
    public Nullable<short> Stone { get; set; }
    public Nullable<short> Pound { get; set; }
    public Nullable<System.DateTime> Date { get; set; }
}
.....
public class WeightViewModel
{
    public IList<AddWeightModel> AddWeightModel { get; set; }
    public Weight Weight { get; set; }
}

Then change your view to accept the view models –

@model WeightViewModel

Finally modify your controller to cope with the change –

public ActionResult RecordCard()
    {
        var UserId = User.Identity.GetUserId();
        var weightModel = from m in db.Weights where m.UserId == UserId select m;
        var viewModel = new WeightViewModel
        {
            Weight = weightModel,
            AddWeightModel = new List<AddWeightModel>(){}
        };
        return View(viewModel);
    }

[HttpPost]
[ValidateAntiForgeryToken]
public ActionResult RecordCard(WeightViewModel viewModel)
{
    Weight Model = viewModel.Weight;
    if (ModelState.IsValid)
    {
        using (WebApplication1Entities db = new WebApplication1Entities())
        {
            Weight weight = new Weight();
            weight.UserId = User.Identity.GetUserId();
            weight.Stone = Model.Stone;
            weight.Pound = Model.Pound;
            weight.Date = System.DateTime.Now;

            db.Weights.Add(Model);
            db.SaveChanges();
        }
    }
    return RedirectToAction("RecordCard");
}